SECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS x 2 y 2 z sx 2 y 2 z 2 2. xyplane. It is sketched in Figure 11.


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1 SECTION 9.1 THREEDIMENSIONAL 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 halfspace 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 threedimensional 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 CauchSchwar 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 CauchSchwar 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 righthand 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 60N 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 36lb 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 TECHarts 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 onethird 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 threedimensional 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 welldefined 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 threedimensional 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 greatcircle 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 50N 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 problemsolving 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
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