SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS x 2 y 2 z sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.

Size: px
Start display at page:

Download "SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11."

Transcription

1 SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities can be rewritten as 2 FIGURE s so the represent the points,, whose distance from the origin is at least 1 and at most 2. But we are also given that 0, so the points lie on or below the -plane. Thus, the given inequalities represent the region that lies between (or on) the spheres and and beneath (or on) the -plane. It is sketched in Figure Eercises 1. Suppose ou start at the origin, move along the -ais a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of our position? 2. Sketch the points (3, 0, 1), 1, 0, 3, 0, 4, 2, and (1, 1, 0) on a single set of coordinate aes. 3. Which of the points P6, 2, 3, Q5, 1, 4, and R0, 3, 8 is closest to the -plane? Which point lies in the -plane? 4. What are the projections of the point (2, 3, 5) on the -, -, and -planes? Draw a rectangular bo with the origin and (2, 3, 5) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the bo. Find the length of the diagonal of the bo. 5. Describe and sketch the surface in 3 represented b the equation (a) What does the equation 4 represent in 2? What does it represent in 3? Illustrate with sketches. (b) What does the equation 3 represent in 3? What does 5 represent? What does the pair of equations 3, 5 represent? In other words, describe the set of points,, such that 3 and 5. Illustrate with a sketch. 7. Find the lengths of the sides of the triangle with vertices A3, 4, 1, B5, 3, 0, and C6, 7, 4. Is ABC a right triangle? Is it an isosceles triangle? 8. Find the distance from 3, 7, 5 to each of the following. (a) The -plane (b) The -plane (c) The -plane (d) The -ais (e) The -ais (f) The -ais 9. Determine whether the points lie on a straight line. (a) A5, 1, 3, B7, 9, 1, C1, 15, 11 (b) K0, 3, 4, L1, 2, 2, M3, 0, Find an equation of the sphere with center 6, 5, 2 and radius s7. Describe its intersection with each of the coordinate planes. 11. Find an equation of the sphere that passes through the point 4, 3, 1 and has center (3, 8, 1). 12. Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3) Show that the equation represents a sphere, and find its center and radius (a) Prove that the midpoint of the line segment from P 1 1, 1, to is 1 P 2 2, 2, ,, (b) Find the lengths of the medians of the triangle with vertices A1, 2, 3, B2, 0, 5, and C4, 1, Find an equation of a sphere if one of its diameters has endpoints 2, 1, 4 and 4, 3, Find equations of the spheres with center 2, 3, 6 that touch (a) the -plane, (b) the -plane, (c) the -plane. 18. Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant Describe in words the region of 3 represented b the equation or inequalit

2 652 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE L P Write inequalities to describe the region. 29. The half-space consisting of all points to the left of the -plane 30. The solid rectangular bo in the first octant bounded b the planes 1, 2, and The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r R 32. The solid upper hemisphere of the sphere of radius 2 centered at the origin 33. The figure shows a line L 1 in space and a second line L 2, which is the projection of L 1 on the -plane. (In other words, the points on L 2 are directl beneath, or above, the points on L 1.) (a) Find the coordinates of the point P on the line L 1. (b) Locate on the diagram the points A, B, and C, where the line L 1 intersects the -plane, the -plane, and the -plane, respectivel Consider the points P such that the distance from P to A1, 5, 3 is twice the distance from P to B6, 2, 2. Show that the set of all such points is a sphere, and find its center and radius. 35. Find an equation of the set of all points equidistant from the points A1, 5, 3 and B6, 2, 2. Describe the set. 36. Find the volume of the solid that lies inside both of the spheres and L 9.2 Vectors A v B C FIGURE 1 Equivalent vectors u D The term vector is used b scientists to indicate a quantit (such as displacement or velocit or force) that has both magnitude and direction. A vector is often represented b an arrow or a directed line segment. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector. We denote a vector b printing a letter in boldface v or b putting an arrow above the letter v l. For instance, suppose a particle moves along a line segment from point A to point B. The corresponding displacement vector v, shown in Figure 1, has initial point (the tail) and terminal point (the tip) and we indicate this b writing AB l A. Notice that the vector CD l B v u has the same length and the same direction as v even though it is in a different position. We sa that u and v are equivalent (or equal) and we write u v. The ero vector, denoted b 0, has length 0. It is the onl vector with no specific direction. Combining Vectors Suppose a particle moves from A to B, so its displacement vector is AB l. Then the particle changes direction and moves from B to C, with displacement vector BC l as in

3 SECTION 9.2 VECTORS 659 So the magnitudes of the tensions are T lb sin 50 tan 32 cos 50 and T 2 T 1 cos 50 cos lb Substituting these values in (5) and (6), we obtain the tension vectors T i j T i j 9.2 Eercises 1. Are the following quantities vectors or scalars? Eplain. (a) The cost of a theater ticket (b) The current in a river (c) The initial flight path from Houston to Dallas (d) The population of the world 2. What is the relationship between the point (4, 7) and the vector 4, 7? Illustrate with a sketch. 3. Name all the equal vectors in the parallelogram shown. A B 6. Cop the vectors in the figure and use them to draw the following vectors. (a) a b (b) a b (c) 2a (d) 1 2 b (e) 2a b (f) b 3a a b E D C 4. Write each combination of vectors as a single vector. (a) PQ l QR l (b) RP l PS l l l (c) QS PS l l l (d) RS SP PQ Q P R S 5. Cop the vectors in the figure and use them to draw the following vectors. (a) u v (b) u v (c) v w (d) w v u u v w 7 10 Find a vector a with representation given b the directed line segment AB l. Draw AB l and the equivalent representation starting at the origin. 7. A1, 1, B3, 4 8. A2, 2, 9. A0, 3, 1, 10. A1, 2, 0, Find the sum of the given vectors and illustrate geometricall , 1, 2, , 2, 13. 1, 0, 1, 0, 0, , 3, 2, a Find, a b, a b, 2a, and 3a 4b. 15. a 4, 3, 16. a 2i 3j, 17. a i 2j k, 18. a 3i 2k, B2, 3, 1 B1, 2, 3 b 6, 2 b i 5j b j 2k b i j k B3, 0 5, 3 1, 0, 3

4 660 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 19. Find a unit vector with the same direction as 8i j 4k. 20. Find a vector that has the same direction as 2, 4, 2 but has length If v lies in the first quadrant and makes an angle 3 with the positive -ais and, find v in component form. 22. If a child pulls a sled through the snow with a force of 50 N eerted at an angle of 38 above the horiontal, find the horiontal and vertical components of the force. 23. Two forces F 1 and F 2 with magnitudes 10 lb and 12 lb act on an object at a point P as shown in the figure. Find the resultant force F acting at P as well as its magnitude and its direction. (Indicate the direction b finding the angle shown in the figure.) F Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocit vector is called speed. Suppose that a wind is blowing from the direction N45W at a speed of 50 kmh. (This means that the direction from which the wind blows is 45 west of the northerl direction.) A pilot is steering a plane in the direction N60E at an airspeed (speed in still air) of 250 kmh. The true course, or track, of the plane is the direction of the resultant of the velocit vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane. 25. A woman walks due west on the deck of a ship at 3 mih. The ship is moving north at a speed of 22 mih. Find the speed and direction of the woman relative to the surface of the water. 26. Ropes 3 m and 5 m in length are fastened to a holida decoration that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52 and 40 with the horiontal. Find the tension in each wire and the magnitude of each tension. 52 v 4 P F m 5 m F 27. A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the clothesline. 28. The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain? (a) Draw the vectors a 3, 2, b 2, 1, and c 7, 1. (b) Show, b means of a sketch, that there are scalars s and t such that c sa tb. (c) Use the sketch to estimate the values of s and t. (d) Find the eact values of s and t. 30. Suppose that a and b are nonero vectors that are not parallel and c is an vector in the plane determined b a and b. Give a geometric argument to show that c can be written as c sa tb for suitable scalars s and t. Then give an argument using components. 31. Suppose a is a three-dimensional unit vector in the first octant that starts at the origin and makes angles of 60 and 72 with the positive - and -aes, respectivel. Epress a in terms of its components. 32. Suppose a vector a makes angles,, and with the positive -, -, and -aes, respectivel. Find the components of a and show that (The numbers cos, cos, and cos are called the direction cosines of a.) 33. If r,, and r 0 0, 0, 0, describe the set of all points,, such that. 34. If r,, r 1 1, 1, and r 2 2, 2, describe the set of all points, such that r r 1 r r 2 k, where. k r 1 r 2 cos 2 cos 2 cos 2 1 r r Figure 16 gives a geometric demonstration of Propert 2 of vectors. Use components to give an algebraic proof of this fact for the case n Prove Propert 5 of vectors algebraicall for the case n 3. Then use similar triangles to give a geometric proof. 37. Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

5 SECTION 9.3 THE DOT PRODUCT Suppose the three coordinate planes are all mirrored and a light ra given b the vector a a 1, a 2, a 3 first strikes the -plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ra is given b b a 1, a 2, a 3. Deduce that, after being reflected b all three mutuall perpendicular mirrors, the resulting ra is parallel to the initial ra. (American space scientists used this principle, together with laser beams and an arra of corner mirrors on the Moon, to calculate ver precisel the distance from Earth to the Moon.) a b 9.3 The Dot Product So far we have added two vectors and multiplied a vector b a scalar. The question arises: Is it possible to multipl two vectors so that their product is a useful quantit? One such product is the dot product, which we consider in this section. Another is the cross product, which is discussed in the net section. Work and the Dot Product P FIGURE 1 F D S R Q An eample of a situation in phsics and engineering where we need to combine two vectors occurs in calculating the work done b a force. In Section 6.5 we defined the work done b a constant force F in moving an object through a distance d as W Fd, but this applies onl when the force is directed along the line of motion of the object. Suppose, however, that the constant force is a vector F PR l pointing in some other direction, as in Figure 1. If the force moves the object from to, then the displacement vector is PQ l P Q D. So here we have two vectors: the force F and the displacement D. The work done b F is defined as the magnitude of the displacement, D, multiplied b the magnitude of the applied force in the direction of the motion, which, from Figure 1, is PS l F cos So the work done b F is defined to be 1 W D ( F cos ) F D cos Notice that work is a scalar quantit; it has no direction. But its value depends on the angle between the force and displacement vectors. We use the epression in Equation 1 to define the dot product of two vectors even when the don t represent force or displacement. Definition The dot product of two nonero vectors a and b is the number a b a b cos where is the angle between a and b, 0. (So is the smaller angle between the vectors when the are drawn with the same initial point.) If either a or b is 0, we define a b 0.

6 666 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 9.3 Eercises 1. Which of the following epressions are meaningful? Which are meaningless? Eplain. (a) a b c (b) a bc (c) a b c (d) a b c (e) a b c (f) b c 2. Find the dot product of two vectors if their lengths are 6 1 and and the angle between them is Find a b. 3.,, the angle between a and b is 6 4., 5. a 5, 0, 2, 6. a s, 2s, 3s, 7. a i 2j 3k, 8. a 4j 3k, 9 10 If u is a unit vector, find u v and u w u w 11. (a) Show that i j j k k i 0. (b) Show that i i j j k k A street vendor sells a hamburgers, b hot dogs, and c soft drinks on a given da. He charges $2 for a hamburger, $1.50 for a hot dog, and $1 for a soft drink. If A a, b, c and P 2, 1.5, 1, what is the meaning of the dot product A P? Find the angle between the vectors. (First find an eact epression and then approimate to the nearest degree.) 13. a 3, 4, 14. a 6, 3, 2, 15. a j k, 3 a 12 a 1 2, 4 b 15 b 8, 3 v b 3, 1, 10 b t, t, 5t b 2i 4j 6k b 5, 12 b 5i 9k b 2, 1, 2 b i 2j 3k a 16. Find, correct to the nearest degree, the three angles of the triangle with the vertices P0, 1, 6, Q2, 1, 3, and R5, 4, 2. w v u 17. Determine whether the given vectors are orthogonal, parallel, or neither. (a) a 5, 3, 7, b 6, 8, 2 (b) a 4, 6, b 3, 2 (c) a i 2j 5k, b 3i 4j k (d) a 2i 6j 4k, b 3i 9j 6k 18. For what values of b are the vectors 6, b, 2 and b, b 2, b orthogonal? 19. Find a unit vector that is orthogonal to both i j and i k. 20. For what values of c is the angle between the vectors 1, 2, 1 and 1, 0, c equal to 60? Find the scalar and vector projections of b onto a. 21. a 2, 3, b 4, a 3, 1, b 2, a 4, 2, 0, b 1, 1, a 2i 3j k, b i 6j 2k 25. Show that the vector orth a b b proj a b is orthogonal to a. (It is called an orthogonal projection of b.) 26. For the vectors in Eercise 22, find orth a b and illustrate b drawing the vectors a, b, proj a b, and orth a b. 27. If a 3, 0, 1, find a vector b such that comp a b Suppose that a and b are nonero vectors. (a) Under what circumstances is comp a b comp b a? (b) Under what circumstances is proj a b proj b a? 29. A constant force with vector representation F 10i 18j 6k moves an object along a straight line from the point 2, 3, 0 to the point 4, 9, 15. Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons. 30. Find the work done b a force of 20 lb acting in the direction N50W in moving an object 4 ft due west. 31. A woman eerts a horiontal force of 25 lb on a crate as she pushes it up a ramp that is 10 ft long and inclined at an angle of 20 above the horiontal. Find the work done on the bo. 32. A wagon is pulled a distance of 100 m along a horiontal path b a constant force of 50 N. The handle of the wagon is held at an angle of 30 above the horiontal. How much work is done? 33. Use a scalar projection to show that the distance from a point P 1 1, 1 to the line a b c 0 is a1 b1 c sa 2 b 2

7 SECTION 9.4 THE CROSS PRODUCT 667 Use this formula to find the distance from the point 2, 3 to the line If r,,, a a 1, a 2, a 3, and b b 1, b 2, b 3, show that the vector equation r a r b 0 represents a sphere, and find its center and radius. 35. Find the angle between a diagonal of a cube and one of its edges. 36. Find the angle between a diagonal of a cube and a diagonal of one of its faces. 37. A molecule of methane, CH 4, is structured with the four hdrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed b the H C H combination; it is the angle between the lines that join the carbon atom to two of the hdrogen atoms. Show that the bond angle is about [ Hint: Take the vertices of the tetrahedron to be the points 1, 0, 0, 0, 1, 0, 0, 0, 1, and 1, 1, 1 as shown in the figure. Then the centroid is. H ( 1 2, 1 2, 1 2 ) ] 38. If, where a, b, and c are all nonero vectors, show that c bisects the angle between a and b. c a b b a 39. Prove Propert 4 of the dot product. Use either the definition of a dot product (considering the cases c 0, c 0, and c 0 separatel) or the component form. 40. Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular. 41. Prove the Cauch-Schwar Inequalit: a b a b 42. The Triangle Inequalit for vectors is a b a b (a) Give a geometric interpretation of the Triangle Inequalit. (b) Use the Cauch-Schwar Inequalit from Eercise 41 to prove the Triangle Inequalit. [Hint: Use the fact that a b 2 a b a b and use Propert 3 of the dot product.] 43. The Parallelogram Law states that H C H H a b 2 a b 2 2 a 2 2 b 2 (a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. (See the hint in Eercise 42.) 9.4 The Cross Product r n The cross product a b of two vectors a and b, unlike the dot product, is a vector. For this reason it is also called the vector product. We will see that a b is useful in geometr because it is perpendicular to both a and b. But we introduce this product b looking at a situation where it arises in phsics and engineering. F Torque and the Cross Product FIGURE 1 r F sin F If we tighten a bolt b appling a force to a wrench as in Figure 1, we produce a turning effect called a torque. The magnitude of the torque depends on two things: The distance from the ais of the bolt to the point where the force is applied. This is r, the length of the position vector r. The scalar component of the force F in the direction perpendicular to r. This is the onl component that can cause a rotation and, from Figure 2, we see that it is F sin FIGURE 2 where is the angle between the vectors r and F.

8 674 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 9.4 Eercises 1. State whether each epression is meaningful. If not, eplain wh. If so, state whether it is a vector or a scalar. (a) a b c (b) a b c (c) a b c (d) a b c (e) a b c d (f) a b c d 2 3 Find and determine whether u v is directed into the page or out of the page u =5 60 u v v =10 4. The figure shows a vector a in the -plane and a vector b in the direction of k. Their lengths are a 3 and (a) Find a b. (b) Use the right-hand rule to decide whether the components of a b are positive, negative, or 0. u =6 150 v =8 b Find the cross product a b and verif that it is orthogonal to both a and b. 7. a 1, 1, 0, 8. a 3, 2, 2, 9. a t, t 2, t 3, 10. a i e t j e t k, 11. a 3i 2j 4k, b 3, 2, 1 b 6, 3, 1 b 1, 2t, 3t 2 b 2i e t j e t k b i 2j 3k 12. If a i 2k and b j k, find a b. Sketch a, b, and a b as vectors starting at the origin. 13. Find two unit vectors orthogonal to both 1, 1, 1 and 0, 4, Find two unit vectors orthogonal to both i j and i j k. 15. Find the area of the parallelogram with vertices A2, 1, B0, 4, C4, 2, and D2, Find the area of the parallelogram with vertices K1, 2, 3, L1, 3, 6, M3, 8, 6, and N3, 7, 3. b (a) Find a vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. a 17. P1, 0, 0, Q0, 2, 0, R0, 0, P2, 0, 3, Q3, 1, 0, R5, 2, 2 5. A biccle pedal is pushed b a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. 60 N P 6. Find the magnitude of the torque about P if a 36-lb force is applied as shown. 4 ft P 19. A wrench 30 cm long lies along the positive -ais and grips a bolt at the origin. A force is applied in the direction 0, 3, 4 at the end of the wrench. Find the magnitude of the force needed to suppl 100 J of torque to the bolt. 20. Let v 5j and let u be a vector with length 3 that starts at the origin and rotates in the -plane. Find the maimum and minimum values of the length of the vector u v. In what direction does u v point? Find the volume of the parallelepiped determined b the vectors a, b, and c. 21. a 6, 3, 1, b 0, 1, 2, 22. a 2i 3j 2k, b i j, c 4, 2, 5 c 2i 3k lb 4 ft Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 23. P1, 1, 1, Q2, 0, 3, R4, 1, 7, S3, 1, P0, 1, 2, Q2, 4, 5, R1, 0, 1, S6, 1, 4

9 DISCOVERY PROJECT THE GEOMETRY OF A TETRAHEDRON Use the scalar triple product to verif that the vectors a 2i 3j k, b i j, and c 7i 3j 2k are coplanar. 26. Use the scalar triple product to determine whether the points P1, 0, 1, Q2, 4, 6, R3, 1, 2, and S6, 2, 8 lie in the same plane. 27. (a) Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is d a b a where a QR l and b QP l. (b) Use the formula in part (a) to find the distance from the point P1, 1, 1 to the line through Q0, 6, 8 and R1, 4, (a) Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is d a b c a b where QR l l l a, b QS, and c QP. (b) Use the formula in part (a) to find the distance from the point P2, 1, 4 to the plane through the points Q1, 0, 0, R0, 2, 0,and S0, 0, Prove that a b a b 2a b. 30. Prove the following formula for the vector triple product: a b c a cb a bc 31. Use Eercise 30 to prove that 32. Prove that 33. Suppose that a 0. (a) If a b a c, does it follow that b c? (b) If a b a c, does it follow that b c? (c) If a b a c and a b a c, does it follow that b c? v 1 a b c b c a c a b 0 v 2 a b c d a c a d v If,, and are noncoplanar vectors, let k 1 k 2 k 3 v 2 v 3 v 1 v 2 v 3 v 3 v 1 v 1 v 2 v 3 v 1 v 2 v 1 v 2 v 3 b c b d (These vectors occur in the stud of crstallograph. Vectors of the form n 1v 1 n 2v 2 n 3v 3, where each n i is an integer, form a lattice for a crstal. Vectors written similarl in terms of k 1, k 2, and k 3 form the reciprocal lattice.) (a) Show that k i is perpendicular to v j if i j. (b) Show that k i v i 1 for i 1, 2, 3. 1 (c) Show that k 1 k 2 k 3. v 1 v 2 v 3 Discover Project The Geometr of a Tetrahedron P A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces, as shown in the figure. 675 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE Stewart 2C3 galles TECH-arts 2C Let v 1, v 2, v 3, and v 4 be vectors with lengths equal to the areas of the faces opposite the vertices P, Q, R, and S, respectivel, and directions perpendicular to the respective faces and pointing outward. Show that S Q R v 1 v 2 v 3 v The volume V of a tetrahedron is one-third the distance from a verte to the opposite face, times the area of that face. (a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S. (b) Find the volume of the tetrahedron whose vertices are P1, 1, 1, Q1, 2, 3, R1, 1, 2, and S3, 1, 2.

10 SECTION 9.5 EQUATIONS OF LINES AND PLANES 683 SOLUTION Since the two lines L 1 and L 2 are skew, the can be viewed as ling on two parallel planes P 1 and P 2. The distance between L 1 and L 2 is the same as the distance between P 1 and P 2, which can be computed as in Eample 9. The common normal vector to both planes must be orthogonal to both v (the direction of ) and (the direction of ). So a normal vector is 1 1, 3, 1 L 1 v 2 2, 1, 4 L 2 i j k n v 1 v i 6j 5k If we put s 0 in the equations of L 2, we get the point 0, 3, 3 on L 2 and so an equation for P 2 is or If we now set t 0 in the equations for L 1, we get the point 1, 2, 4 on P 1. So the distance between L 1 and L 2 is the same as the distance from 1, 2, 4 to B Formula 8, this distance is D s s Eercises 1. Determine whether each statement is true or false. (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel. 2 5 Find a vector equation and parametric equations for the line. 2. The line through the point 1, 0, 3 and parallel to the vector 2i 4j 5k 3. The line through the point 2, 4, 10 and parallel to the vector 3, 1, 8 4. The line through the origin and parallel to the line 2t, 1 t, 4 3t 5. The line through the point (1, 0, 6) and perpendicular to the plane Find parametric equations and smmetric equations for the line. 6. The line through the origin and the point 1, 2, 3 7. The line through the points 3, 1, 1 and 3, 2, 6 8. The line through the points 1, 0, 5 and 4, 3, 3 9. The line through the points (0, 1 2, 1) and 2, 1, The line of intersection of the planes 1 and Show that the line through the points 2, 1, 5 and 8, 8, 7 is parallel to the line through the points 4, 2, 6 and 8, 8, Show that the line through the points 0, 1, 1 and 1, 1, 6 is perpendicular to the line through the points 4, 2, 1 and 1, 6, (a) Find smmetric equations for the line that passes through the point 0, 2, 1 and is parallel to the line with parametric equations 1 2t, 3t, 5 7t. (b) Find the points in which the required line in part (a) intersects the coordinate planes.

11 684 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 14. (a) Find parametric equations for the line through 5, 1, 0 that is perpendicular to the plane 2 1. (b) In what points does this line intersect the coordinate planes? Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. If the intersect, find the point of intersection. L : L 2 : L 1 : 1, L 2 : L 1 : 6t, 1 9t, 3t L 2 : 1 2s, 4 3s, s 18. L 1 : 1 t, 2 t, 3t L 2 : 2 s, 1 2s, 4 s Find an equation of the plane. 19. The plane through the point 6, 3, 2 and perpendicular to the vector 2, 1, The plane through the point 4, 0, 3 and with normal vector j 2k 21. The plane through the origin and parallel to the plane The plane that contains the line 3 2t, t, 8 t and is parallel to the plane The plane through the points 0, 1, 1, 1, 0, 1, and 1, 1, The plane through the origin and the points 2, 4, 6 and 5, 1, The plane that passes through the point 6, 0, 2 and contains the line 4 2t, 3 5t, 7 4t 26. The plane that passes through the point 1, 1, 1 and contains the line with smmetric equations The plane that passes through the point 1, 2, 1 and contains the line of intersection of the planes 2 and The plane that passes through the line of intersection of the planes 1 and 2 3 and is perpendicular to the plane Find the point at which the line intersects the given plane t, 1, t; t, t, 1 t; Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them , , , , 35. (a) Find smmetric equations for the line of intersection of the planes 2 and (b) Find the angle between these planes. 36. Find an equation for the plane consisting of all points that are equidistant from the points 4, 2, 1 and 2, 4, Find an equation of the plane with -intercept a, -intercept b, and -intercept c. 38. (a) Find the point at which the given lines intersect: and r 2, 0, 2 s1, 1, 0 (b) Find an equation of the plane that contains these lines. 39. Find parametric equations for the line through the point 0, 1, 2 that is parallel to the plane 2 and perpendicular to the line 1 t, 1 t, 2t. 40. Find parametric equations for the line through the point 0, 1, 2 that is perpendicular to the line 1 t, 1 t, 2t and intersects this line. 41. Which of the following four planes are parallel? Are an of them identical? P 1: P 2: P 3: P 4: Which of the following four lines are parallel? Are an of them identical? L 1: 1 t, t, 2 5t Use the formula in Eercise 27 in Section 9.4 to find the distance from the point to the given line , 2, 3; 2 t, 2 3t, 5t 44. 1, 0, 1; 5 t, 3t, 1 2t Find the distance from the point to the given plane , 8, 5, 46. 3, 2, 7, 1 r 1, 1, 0 t1, 1, 2 L 2: L 3: 1 t, 4 t, 1 t L 4: r 2, 1, 3 t2, 2,

12 SECTION 9.6 FUNCTIONS AND SURFACES Find the distance between the given parallel planes , , Show that the distance between the parallel planes a b c d 1 0 and a b c d 2 0 is D d1 d2 sa 2 b 2 c Find equations of the planes that are parallel to the plane and two units awa from it. 51. Show that the lines with smmetric equations and are skew, and find the distance between these lines. 52. Find the distance between the skew lines with parametric equations 1 t, 1 6t, 2t, and 1 2s, 5 15s, 2 6s. 53. If a, b, and c are not all 0, show that the equation a b c d 0 represents a plane and a, b, c is a normal vector to the plane. Hint: Suppose a 0 and rewrite the equation in the form a d a b 0 c Give a geometric description of each famil of planes. (a) c (b) c 1 (c) cos sin Functions and Surfaces In this section we take a first look at functions of two variables and their graphs, which are surfaces in three-dimensional space. We will give a much more thorough treatment of such functions in Chapter 11. Functions of Two Variables The temperature T at a point on the surface of the earth at an given time depends on the longitude and latitude of the point. We can think of T as being a function of the two variables and, or as a function of the pair,. We indicate this functional dependence b writing T f,. The volume V of a circular clinder depends on its radius r and its height h. In fact, we know that V r 2 h. We sa that V is a function of r and h, and we write Vr, h r 2 h. Definition A function f of two variables is a rule that assigns to each ordered pair of real numbers, in a set D a unique real number denoted b f,. The set D is the domain of f and its range is the set of values that f takes on, that is, f,, D. We often write f, to make eplicit the value taken on b f at the general point,. The variables and are independent variables and is the dependent variable. [Compare this with the notation f for functions of a single variable.] The domain is a subset of 2, the -plane. We can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. If a function f is given b a formula and no domain is specified, then the domain of f is understood to be the set of all pairs, for which the given epression is a well-defined real number.

13 692 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 9 Classif the quadric surface SOLUTION B completing the square we rewrite the equation as (3, 1, 0) Comparing this equation with Table 2, we see that it represents an elliptic paraboloid. Here, however, the ais of the paraboloid is parallel to the -ais, and it has been shifted so that its verte is the point 3, 1, 0. The traces in the plane k k 1 are the ellipses k 1 k FIGURE 13 The trace in the -plane is the parabola with equation 1 3 2, 0. The paraboloid is sketched in Figure Eercises 1. In Eample 3 we considered the function h f v, t, where h is the height of waves produced b wind at speed v for a time t. Use Table 1 to answer the following questions. (a) What is the value of f 40, 15? What is its meaning? (b) What is the meaning of the function h f 30, t? Describe the behavior of this function. (c) What is the meaning of the function h f v, 30? Describe the behavior of this function. 2. The figure shows vertical traces for a function f,. Which one of the graphs I IV has these traces? Eplain. k=1 k=_1 3. Let f, 2 e 3. (a) Evaluate f 2, 0. (b) Find the domain of f. (c) Find the range of f. 4. Let f, ln 1. (a) Evaluate f 1, 1. (b) Evaluate f e, 1. (c) Find and sketch the domain of f. (d) Find the range of f. 5 8 Find and sketch the domain of the function. 5. f, s 6. f, s s _2 0 2 _ f, s f, s ln4 2 2 _ Sketch the graph of the function. I Traces in =k II Traces in =k 9. f, f, f, 1 2 f, f, sin III IV 14. (a) Find the traces of the function f, 2 2 in the planes k, k, and k. Use these traces to sketch the graph. (b) Sketch the graph of t, 2 2. How is it related to the graph of f? (c) Sketch the graph of h, How is it related to the graph of t? 15. Match the function with its graph (labeled I VI). Give reasons for our choices. (a) f, (b) f,

14 SECTION 9.6 FUNCTIONS AND SURFACES 693 (c) (e) (d) (f) I II 1 f, f, 2 III IV f, f, sin( ) the graph of the hperboloid of one sheet in Table 2. (b) If we change the equation in part (a) to , how is the graph affected? (c) What if we change the equation in part (a) to ? 26. (a) Find and identif the traces of the quadric surface and eplain wh the graph looks like the graph of the hperboloid of two sheets in Table 2. (b) If the equation in part (a) is changed to , what happens to the graph? Sketch the new graph. V VI ; Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the peaks and valles. Would ou sa the function has a maimum value? Can ou identif an points on the graph that ou might consider to be local maimum points? What about local minimum points? 27. f, f, e Use traces to sketch the graph of the function f, s f, f, Use traces to sketch the surface Classif the surface b comparing with one of the standard forms in Table 2. Then sketch its graph (a) What does the equation represent as a curve in 2? (b) What does it represent as a surface in 3? (c) What does the equation represent? 24. (a) Identif the traces of the surface (b) Sketch the surface. (c) Sketch the graphs of the functions f, s 2 2 and t, s (a) Find and identif the traces of the quadric surface and eplain wh the graph looks like ; Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both and become large? What happens as, approaches the origin? 29. f, 30. f, ; 31. Graph the surfaces 2 2 and 1 2 on a common screen using the domain 1.2, 1.2 and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the -plane is an ellipse. 32. Show that the curve of intersection of the surfaces and lies in a plane. 33. Show that if the point a, b, c lies on the hperbolic paraboloid 2 2, then the lines with parametric equations a t, b t, c 2b at and a t, b t, c 2b at both lie entirel on this paraboloid. (This shows that the hperbolic paraboloid is what is called a ruled surface; that is, it can be generated b the motion of a straight line. In fact, this eercise shows that through each point on the hperbolic paraboloid there are two generating lines. The onl other quadric surfaces that are ruled surfaces are clinders, cones, and hperboloids of one sheet.) 34. Find an equation for the surface consisting of all points P for which the distance from P to the -ais is twice the distance from P to the -plane. Identif the surface.

15 698 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE EXAMPLE 8 Use a computer to draw a picture of the solid that remains when a hole of radius 3 is drilled through the center of a sphere of radius 4. SOLUTION To keep the equations simple, let s choose the coordinate sstem so that the center of the sphere is at the origin and the ais of the clinder that forms the hole is the -ais. We could use either clindrical or spherical coordinates to describe the solid, but the description is somewhat simpler if we use clindrical coordinates. Then the equation of the clinder is r 3 and the equation of the sphere is , or r The points in the solid lie outside the clinder and inside the sphere, so the satisf the inequalities 3 r s16 2 To ensure that the computer graphs onl the appropriate parts of these surfaces, we find where the intersect b solving the equations r 3 and r s16 2 : s16 2 3? ? 2 7? s7 The solid lies between s7 and s7, so we ask the computer to graph the surfaces with the following equations and domains: r 3 r s s7 s7 s7 s7 The resulting picture, shown in Figure 11, is eactl what we want. Most three-dimensional graphing programs can graph surfaces whose equations are given in clindrical or spherical coordinates. As Eample 8 demonstrates, this is often the most convenient wa of drawing a solid. FIGURE Eercises 1. What are clindrical coordinates? For what tpes of surfaces do the provide convenient descriptions? 2. What are spherical coordinates? For what tpes of surfaces do the provide convenient descriptions? 3 4 Plot the point whose clindrical coordinates are given. Then find the rectangular coordinates of the point. 3. (a) 3, 2, 1 (b) 4, 3, 5 4. (a) 1,, e (b) 5, 6, Change from rectangular to clindrical coordinates. 5. (a) 1, 1, 4 (b) (1, s3, 2) 6. (a) 3, 3, 2 (b) 3, 4, Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. 7. (a) 1, 0, 0 (b) 2, 3, 4 8. (a) 5,, 2 (b) 2, 4, 3

16 LABORATORY PROJECT FAMILIES OF SURFACES Change from rectangular to spherical coordinates. 9. (a) 3, 0, 0 (b) 0, 2, (a) (1, s3, 2) (b) 0, 0, Describe in words the surface whose equation is given. 11. r Identif the surface whose equation is given. 15. r r 2 cos r r Write the equation (a) in clindrical coordinates and (b) in spherical coordinates Sketch the solid described b the given inequalities. 25. r 2 2 r , r , 0 6, , sin 2 2 cos sec 29. A clindrical shell is 20 cm long, with inner radius 6 cm and outer radius 7 cm. Write inequalities that describe the shell in an appropriate coordinate sstem. Eplain how ou have positioned the coordinate sstem with respect to the shell. 30. (a) Find inequalities that describe a hollow ball with diameter 30 cm and thickness 0.5 cm. Eplain how ou have positioned the coordinate sstem that ou have chosen. (b) Suppose the ball is cut in half. Write inequalities that describe one of the halves. 31. A solid lies above the cone s 2 2 and below the sphere Write a description of the solid in terms of inequalities involving spherical coordinates. ; 32. Use a graphing device to draw the solid enclosed b the paraboloids 2 2 and ; 33. Use a graphing device to draw a silo consisting of a clinder with radius 3 and height 10 surmounted b a hemisphere. 34. The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates,, as follows. We take the origin to be the center of the Earth and the positive -ais to pass through the North Pole. The positive -ais passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of P is and the longitude is. Find the great-circle distance from Los Angeles (lat N, long W) to Montréal (lat N, long W). Take the radius of the Earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.) Laborator Project ; Families of Surfaces In this project ou will discover the interesting shapes that members of families of surfaces can take. You will also see how the shape of the surface evolves as ou var the constants. 1. Use a computer to investigate the famil of functions f, a 2 b 2 e 2 2 How does the shape of the graph depend on the numbers a and b? 2. Use a computer to investigate the famil of surfaces 2 2 c. In particular, ou should determine the transitional values of c for which the surface changes from one tpe of quadric surface to another. 3. Members of the famil of surfaces given in spherical coordinates b the equation sin m sin n have been suggested as models for tumors and have been called bump spheres and wrinkled spheres. Use a computer to investigate this famil of surfaces, assuming that m and n are positive integers. What roles do the values of m and n pla in the shape of the surface?

17 700 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 9 Review CONCEPT CHECK 1. What is the difference between a vector and a scalar? 2. How do ou add two vectors geometricall? How do ou add them algebraicall? 3. If a is a vector and c is a scalar, how is ca related to a geometricall? How do ou find ca algebraicall? 4. How do ou find the vector from one point to another? 5. How do ou find the dot product a b of two vectors if ou know their lengths and the angle between them? What if ou know their components? 6. How are dot products useful? 7. Write epressions for the scalar and vector projections of b onto a. Illustrate with diagrams. 8. How do ou find the cross product a b of two vectors if ou know their lengths and the angle between them? What if ou know their components? 9. How are cross products useful? 10. (a) How do ou find the area of the parallelogram determined b a and b? (b) How do ou find the volume of the parallelepiped determined b a, b, and c? 11. How do ou find a vector perpendicular to a plane? 12. How do ou find the angle between two intersecting planes? 13. Write a vector equation, parametric equations, and smmetric equations for a line. 14. Write a vector equation and a scalar equation for a plane. 15. (a) How do ou tell if two vectors are parallel? (b) How do ou tell if two vectors are perpendicular? (c) How do ou tell if two planes are parallel? 16. (a) Describe a method for determining whether three points P, Q, and R lie on the same line. (b) Describe a method for determining whether four points P, Q, R, and S lie in the same plane. 17. (a) How do ou find the distance from a point to a line? (b) How do ou find the distance from a point to a plane? (c) How do ou find the distance between two lines? 18. How do ou sketch the graph of a function of two variables? 19. Write equations in standard form of the si tpes of quadric surfaces. 20. (a) Write the equations for converting from clindrical to rectangular coordinates. In what situation would ou use clindrical coordinates? (b) Write the equations for converting from spherical to rectangular coordinates. In what situation would ou use spherical coordinates? TRUE FALSE QUIZ Determine whether the statement is true or false. If it is true, eplain wh. If it is false, eplain wh or give an eample that disproves the statement. 1. For an vectors u and v in, u v v u. 2. For an vectors u and v in, u v v u. 3. For an vectors u and v in,. 4. For an vectors u and v in V 3 and an scalar k, ku v ku v. 5. For an vectors u and v in V 3 and an scalar k, ku v ku v. V 3 V 3 V 3 6. For an vectors u, v, and w in V 3, u v w u w v w. u v v u 7. For an vectors u, v, and w in V 3, u v w u v w. 8. For an vectors u, v, and w in V 3, u v w u v w. 9. For an vectors u and v in, u v u For an vectors u and v in, u v v u v. 11. The cross product of two unit vectors is a unit vector. 12. A linear equation A B C D 0 represents a line in space. {,, 2 2 1} 13. The set of points is a circle. V 3 V If u u 1, u 2 and v v 1, v 2, then u v u 1v 1, u 2v 2.

18 CHAPTER 9 REVIEW 701 EXERCISES 1. (a) Find an equation of the sphere that passes through the point 6, 2, 3 and has center 1, 2, 1. (b) Find the curve in which this sphere intersects the -plane. (c) Find the center and radius of the sphere Cop the vectors in the figure and use them to draw each of the following vectors. (a) a b (b) a b (c) 1 2 a (d) 2a b 11. (a) Find a vector perpendicular to the plane through the points A1, 0, 0, B2, 0, 1, and C1, 4, 3. (b) Find the area of triangle ABC. 12. A constant force F 3i 5j 10k moves an object along the line segment from 1, 0, 2 to 5, 3, 8. Find the work done if the distance is measured in meters and the force in newtons. 13. A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of 255 N is needed, find the magnitude of the force in each rope. a b N 3. If u and v are the vectors shown in the figure, find u v and. Is u v directed into the page or out of it? u v v =3 45 u =2 4. Calculate the given quantit if a i j 2k b 3i 2j k c j 5k (a) 2a 3b (b) (c) a b (d) a b (e) b c (f ) a b c (g) c c (h) a b c (i) comp a b (j) proj a b (k) The angle between a and b (correct to the nearest degree) 5. Find the values of such that the vectors 3, 2, and 2, 4, are orthogonal. 6. Find two unit vectors that are orthogonal to both j 2k and i 2j 3k. 7. Suppose that u v w 2. Find (a) u v w (b) u w v (c) v u w (d) u v v 8. Show that if a, b, and c are in, then a b b c c a a b c 2 9. Find the acute angle between two diagonals of a cube. 10. Given the points A1, 0, 1, B2, 3, 0, C1, 1, 4, and D0, 3, 2, find the volume of the parallelepiped with adjacent edges AB, AC, and AD. V 3 b 14. Find the magnitude of the torque about P if a 50-N force is applied as shown. 40 cm Find parametric equations for the line that satisfies the given conditions. 15. Passing through 1, 2, 4 and in the direction of v 2i j 3k 16. Passing through 6, 1, 0 and 2, 3, 5 P 50 N Passing through 1, 0, 1 and parallel to the line with parametric equations 4t, 1 3t, 2 5t Find an equation of the plane that satisfies the given conditions. 18. Passing through 4, 1, 1 and with normal vector 2, 6, Passing through 4, 1, 2 and parallel to the plane Passing through 1, 2, 0, 2, 0, 1, and 5, 3, Passing through the line of intersection of the planes 1 and 2 3 and perpendicular to the plane 2 1

19 702 CHAPTER 9 VECTORS AND THE GEOMETRY OF SPACE 22. Find the point in which the line with parametric equations 2 t, 1 3t, 4t intersects the plane Determine whether the lines given b the smmetric equations and are parallel, skew, or intersecting. 24. (a) Show that the planes 1 and are neither parallel nor perpendicular. (b) Find, correct to the nearest degree, the angle between these planes. 25. Find the distance between the planes and Find the distance from the origin to the line 1 t, 2 t, 1 2t Find and sketch the domain of the function. 27. f, ln f, ssin Sketch the graph of the function. 29. f, f, cos 31. f, f, s Identif and sketch the graph of each surface The clindrical coordinates of a point are 2, 6, 2. Find the rectangular and spherical coordinates of the point. 38. The rectangular coordinates of a point are 2, 2, 1. Find the clindrical and spherical coordinates of the point. 39. The spherical coordinates of a point are 4, 3, 6. Find the rectangular and clindrical coordinates of the point. 40. Identif the surfaces whose equations are given. (a) (b) Write the equation in clindrical coordinates and in spherical coordinates The parabola 4 2, 0 is rotated about the -ais. Write an equation of the resulting surface in clindrical coordinates. 44. Sketch the solid consisting of all points with spherical coordinates,, such that 0, 0 6, and 0. 2 cos 4 2

20 1m 1 m FIGURE FOR PROBLEM 1 N W Focus on Problem Solving F FIGURE FOR PROBLEM 5 1 m 1. Each edge of a cubical bo has length 1 m. The bo contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the bo. Thus, the balls are tightl packed in the bo. (See the figure.) Find r. (If ou have trouble with this problem, read about the problem-solving strateg entitled Use analog on page 88.) 2. Let B be a solid bo with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Epress the volume of S in terms of L, W, and H. 3. Let L be the line of intersection of the planes c c and c c 1, where c is a real number. (a) Find smmetric equations for L. (b) As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersection of S with the horiontal plane t (the trace of S in the plane t). (c) Find the volume of the solid bounded b S and the planes 0 and A plane is capable of fling at a speed of 180 kmh in still air. The pilot takes off from an airfield and heads due north according to the plane s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actuall traveled 80 km at an angle 5 east of north. (a) What is the wind velocit? (b) In what direction should the pilot have headed to reach the intended destination? 5. Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block s descent down the plane is slowed b friction; if is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where W mt ( t is the acceleration due to gravit); the normal force N (the normal component of the reactionar force of the plane on the block), where N n ; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and is increased, F must also increase until ultimatel F reaches its maimum, beond which the block begins to slide. At this angle s, it has been observed that is proportional to. Thus, when is maimal, we can sa that F F n F sn, where s is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N F W 0 and deduce that s tan s. (b) Suppose that, for s, an additional outside force H is applied to the block, horiontall from the left, and let H h. If h is small, the block ma still slide down the plane; if h is large enough, the block will move up the plane. Let h min be the smallest value of h that allows the block to remain motionless (so that F is maimal). B choosing the coordinate aes so that F lies along the -ais, resolve each force into components parallel and perpendicular to the inclined plane and show that h min sin mt cos n and h min cos sn mt sin (c) Show that h min mt tan s Does this equation seem reasonable? Does it make sense for s? As? Eplain. (d) Let h ma be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that h ma mt tan s Does this equation seem reasonable? Eplain. l

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

Click here for answers.

Click here for answers. CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

More information

Two vectors are equal if they have the same length and direction. They do not

Two vectors are equal if they have the same length and direction. They do not Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

More information

THE PARABOLA section. Developing the Equation

THE PARABOLA section. Developing the Equation 80 (-0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

THE PARABOLA 13.2. section

THE PARABOLA 13.2. section 698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

13.4 THE CROSS PRODUCT

13.4 THE CROSS PRODUCT 710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

Applications of Trigonometry

Applications of Trigonometry 5144_Demana_Ch06pp501-566 01/11/06 9:31 PM Page 501 CHAPTER 6 Applications of Trigonometr 6.1 Vectors in the Plane 6. Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5

More information

LINEAR FUNCTIONS OF 2 VARIABLES

LINEAR FUNCTIONS OF 2 VARIABLES CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

More information

SECTION 7-4 Algebraic Vectors

SECTION 7-4 Algebraic Vectors 7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors

More information

CHALLENGE PROBLEMS: CHAPTER 10. Click here for solutions. Click here for answers.

CHALLENGE PROBLEMS: CHAPTER 10. Click here for solutions. Click here for answers. CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 0 A Click here for answers. S Click here for solutions. m m m FIGURE FOR PROBLEM N W F FIGURE FOR PROBLEM 5. Each edge of a cubical bo has length m. The bo

More information

a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp

a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION

More information

Mathematics Notes for Class 12 chapter 10. Vector Algebra

Mathematics Notes for Class 12 chapter 10. Vector Algebra 1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

More information

SECTION 9-1 Conic Sections; Parabola

SECTION 9-1 Conic Sections; Parabola 66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

3 Rectangular Coordinate System and Graphs

3 Rectangular Coordinate System and Graphs 060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter

More information

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM MA6-A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,

More information

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

More information

SL Calculus Practice Problems

SL Calculus Practice Problems Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

More information

REVIEW OF CONIC SECTIONS

REVIEW OF CONIC SECTIONS REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

More information

8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections 8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

Sandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem.

Sandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem. Sandia High School Geometry Second Semester FINL EXM Name: Mark the letter to the single, correct (or most accurate) answer to each problem.. What is the value of in the triangle on the right?.. 6. D.

More information

Section 7-2 Ellipse. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEFINITION

Section 7-2 Ellipse. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEFINITION 7- Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1 Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50 Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

More information

DIFFERENTIATION OPTIMIZATION PROBLEMS

DIFFERENTIATION OPTIMIZATION PROBLEMS DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

COMPONENTS OF VECTORS

COMPONENTS OF VECTORS COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two

More information

Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Exercises. For students who are entering. Pre-Calculus Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Chapter 3A - Rectangular Coordinate System

Chapter 3A - Rectangular Coordinate System - Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been

More information

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN

COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject

More information

Similar Right Triangles

Similar Right Triangles 9.1 Similar Right Triangles Goals p Solve problems involving similar right triangles formed b the altitude drawn to the hpotenuse of a right triangle. p Use a geometric mean to solve problems. THEOREM

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image. Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical

More information

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

More information

Translating Points. Subtract 2 from the y-coordinates

Translating Points. Subtract 2 from the y-coordinates CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

More information

Geometry and Measurement

Geometry and Measurement The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

More information

There are good hints and notes in the answers to the following, but struggle first before peeking at those!

There are good hints and notes in the answers to the following, but struggle first before peeking at those! Integration Worksheet - Using the Definite Integral Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the various types of problems. Important:

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

SECTION 2-2 Straight Lines

SECTION 2-2 Straight Lines - Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

More information

Anytime plan TalkMore plan

Anytime plan TalkMore plan CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

Section 11.4: Equations of Lines and Planes

Section 11.4: Equations of Lines and Planes Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. Chapter 2. Chapter 3 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

Concepts in Calculus III

Concepts in Calculus III Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

Winter 2012 Math 255. Review Sheet for First Midterm Exam Solutions

Winter 2012 Math 255. Review Sheet for First Midterm Exam Solutions Winter 1 Math 55 Review Sheet for First Midterm Exam Solutions 1. Describe the motion of a particle with position x,y): a) x = sin π t, y = cosπt, 1 t, and b) x = cost, y = tant, t π/4. a) Using the general

More information

Chapter 3 Vectors 3.1 Vector Analysis Introduction to Vectors Properties of Vectors Cartesian Coordinate System...

Chapter 3 Vectors 3.1 Vector Analysis Introduction to Vectors Properties of Vectors Cartesian Coordinate System... Chapter 3 Vectors 3.1 Vector Analsis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Cartesian Coordinate Sstem... 5 3.2.1 Cartesian Coordinates... 6 3.3 Application of Vectors...

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

Lines and Planes 1. x(t) = at + b y(t) = ct + d

Lines and Planes 1. x(t) = at + b y(t) = ct + d 1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b

More information

Addition and Subtraction of Vectors

Addition and Subtraction of Vectors ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

More information

2014 2015 Geometry B Exam Review

2014 2015 Geometry B Exam Review Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

MA261-A Calculus III 2006 Fall Homework 2 Solutions Due 9/13/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 2 Solutions Due 9/13/2006 8:00AM MA6-A Calculus III 006 Fall Homework Solutions Due 9/3/006 8:00AM 93 #6 Find a b, where a hs; s; 3si and b ht; t; 5ti a b (s) (t) + (s) ( 93 #8 Find a b, where a 4j 3k and b i + 4j + 6k t) + (3s) (5t)

More information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

STRESS TRANSFORMATION AND MOHR S CIRCLE

STRESS TRANSFORMATION AND MOHR S CIRCLE Chapter 5 STRESS TRANSFORMATION AND MOHR S CIRCLE 5.1 Stress Transformations and Mohr s Circle We have now shown that, in the absence of bod moments, there are si components of the stress tensor at a material

More information

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point. 6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

More information