Question Details SEssCalc [ ] Question Details SEssCalc [ ] Question Details SEssCalc
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1 REVIEW FOR THE EXAM 1 MATH 203 F 2014 ( ) Question Question Details SEssCalc [ ] Write the equation of the sphere in standard form. x 2 + y 2 + z 2 + 4x 2y 6z = 2 Find its center and radius. center radius 2. Question Details SEssCalc [ ] Write the equation of the sphere in standard form. 2x 2 + 2y 2 + 2z 2 = 4x 20z + 1 Find its center and radius. center radius 3. Question Details SEssCalc [ ] Describe the surface in 3 represented by the equation x + y = 8. Page 1 of 19
2 This is the set {(x, 8 x, z) x, z } which is a vertical plane that intersects the xyplane in the line y = 8 x, z = 0. This is the set {(x, 8 x, z) x, z } which is a vertical plane that intersects the xzplane in the line y = 8 x, z = 0. This is the set {(x, 8 x, z) x, z } which is a horizontal plane that intersects the xyplane in the line y = 8 x, z = 0. This is the set {(x, 8 x, z) x, z } which is a horizontal plane that intersects the xzplane in the line y = 8 x, z = 0. This is the set {(x, y, 8 x y) x, y } which is a vertical plane that intersects the xyplane in the line y = 8 x, z = 0. Sketch the surface. Page 2 of 19
3 The equation x + y = 8 represents the set of all points in 3 whose x and ycoordinates have a sum of 8, or equivalently where y = 8 x. This is the set {(x, 8 x, z) x, z } which is a vertical plane that intersects the xyplane in the line y = 8 x, z = Question Details SEssCalc [ ] Consider the point. (1, 4, 6) What is the projection of the point on the xyplane? What is the projection of the point on the yzplane? Page 3 of 19
4 What is the projection of the point on the xzplane? Draw a rectangular box with the origin and (1, 4, 6) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box. Page 4 of 19
5 5. Question Details SEssCalc [ ] Describe in words the region of 3 represented by the inequality. 0 z 5 The inequality 0 z 5 represents all points Select on or between the Select horizontal planes z = 0 (the? xy plane) and z = 5. The inequality 0 z 5 represents all points on or between the horizontal planes z = 0 (the xyplane) and z = Question Details SEssCalc [ ] Find a + b, 2a + 3b, a, and a b. a = 4, 3, b = 4, 3 a + b = 2a + 3b = a = a b = Page 5 of 19
6 7. Question Details SEssCalc [ ] Find a + b, 2a + 3b, a, and a b. a = 2i 4j + 3k, b = 2j k a + b = 2a + 3b = a = a b = 8. Question Details SEssCalc [ ] Find a vector that has the same direction as 2, 6, 2 but has length 6. 2, 6, 2 = ( 2) = 44 = 2 11, so a unit vector in the direction of 2, 6, 2 is u = 2, 6, A vector in the same direction but with length 6 is 6u = 6 2, 6, 2 = 2, 6, Question Details SEssCalc XP. [ ] Find a unit vector that has the same direction as the given vector. 4, 8, 8 4, 8, 8 = ( 4) = = 12, so u = 1 4, 8, 8 = 1 144, 2, Page 6 of 19
7 10. Question Details SEssCalc [ ] If a = 5, 0, 1, find a vector b such that comp a b = 2. b = 11. Question Details SEssCalc [ ] Find the scalar and vector projections of b onto a. a = i + j + k, b = i j + k comp a b = proj a b = 12. Question Details SEssCalc [ ] Find the acute angle between the lines. Round your answer to the nearest degree. 3x y = 5, 2x + y = 8 45 The line 3x y = 5 y = 3x 5 has slope 3, so a vector parallel to the line is a = 1, 3. The line 2x + y = 8 y = 2x + 8 has slope 2, so a vector parallel to the line is b = 1, 2. The angle between the lines is the same as the angle θ between the vectors. Here we have b = ( 2) 2 = 5, so cos θ = a b 5 5 = =. Thus θ = cos 135, and the acute a b angle between the lines is = 45. a b = (1)(1) + (3)( 2) = 5, a = =, and 13. Question Details SEssCalc [ ] Find a unit vector that is orthogonal to both i + j and i + k. Page 7 of 19
8 14. Question Details SEssCalc [ ] Use vectors to decide whether the triangle with vertices Yes, it is rightangled. No, it is not rightangled. P(0, 4, 3), Q(1, 1, 5), and R(5, 3, 6) is rightangled. QP = 1, 3, 2, QR = 4, 2, 1, and QP QR = = 0. Thus QP and QR are orthogonal, so the angle of the triangle at vertex Q is a right angle. 15. Question Details SEssCalc [ ] Determine whether the given vectors are orthogonal, parallel, or neither. (a) u = 3, 3, 6, orthogonal parallel neither v = 4, 4, 8 (b) u = i j + 3k, orthogonal parallel neither v = 3i j + k (c) u = a, b, c, v = b, a, 0 parallel neither orthogonal Page 8 of 19
9 16. Question Details SEssCalc [ ] If u is a unit vector, find u v and u w. (Assume v and w are also unit vectors.) u v = 1/2 u w = 1/2 u, v, and w are all unit vectors, so the triangle is an equilateral triangle. Thus the angle between u and v is 60 and u v = u v cos 60 = (1)(1) 1 = 1. If w is moved so it has the same initial point as u, we can see that the angle 2 2 between them is 120 and we have u w = u w cos 120 = (1)(1) 1 = Page 9 of 19
10 17. Question Details SEssCalc [ ] Find the area of the parallelogram with vertices 16 A( 3, 5), B( 1, 8), C(3, 6), and D(1, 3). By plotting the vertices, we can see that the parallelogram is determined by the vectors AB = 2, 3 and AD = 4, 2. We know that the area of the parallelogram determined by two vectors is equal to the length of the cross product of these vectors. In order to compute the cross product, we consider the vector AB as the threedimensional vector 2, 3, 0 (and similarly for AD), and then the area of parallelogram ABCD is i j k AB AD = = (0)i (0)j + ( 4 12)k = 16k = Page 10 of 19
11 18. Question Details SEssCalc [ ] Consider the points below. P(1, 0, 1), Q( 2, 1, 3), R(4, 2, 5) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR. (a) Because the plane through P, Q, and R contains the vectors PQ and PR, a vector orthogonal to both of these vectors (such as their cross product) is also orthogonal to the plane. Here PQ = 3, 1, 2 and PR = 3, 2, 4, so PQ PR = (1)(4) (2)(2), (3)(2) ( 3)(4), ( 3)(2) (3)(1) = 0, 18, 9 Therefore, 0, 18, 9 or any nonzero scalar multiple thereof, such as 0, 18, 9 is orthogonal to the plane through P, Q, and R. (b) Note that the area of the triangle determined by P, Q, and R is equal to half of the area of the parallelogram determined by the three points. From part (a), the area of the parallelogram is PQ PR = 0, 18, 9 = = 405 = 9 5, so the area of the triangle is = Question Details SEssCalc [ ] Consider the points below. P( 1, 2, 1), Q(0, 6, 3), R(5, 3, 1) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR. Page 11 of 19
12 20. Question Details SEssCalc [ ] Find the volume of the parallelepiped determined by the vectors a, b, and c. a = 1, 5, 4, b = 1, 1, 5, c = 3, 1, 3 72 cubic units Recalling that the volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product, V = a (b c), one obtains a (b c) = = = 1 (3 5) 5 ( 3 15) + 4 ( 1 3) = Thus the volume of the parallelepiped is 72 cubic units. 21. Question Details SEssCalc [ ] Use the scalar triple product to determine if the vectors Yes, they are coplanar. No, they are not coplanar. u = i + 5j 2k, v = 4i j, and w = 8i + 14j 6k are coplanar. 22. Question Details SEssCalc [ ] Use the scalar triple product to determine whether the points same plane. Yes, they lie in the same plane. No, they do not lie in the same plane. A(1, 2, 3), B(4, 3, 7), C(6, 1, 2), and D(3, 6, 2) lie in the Page 12 of 19
13 23. Question Details SEssCalc MI. [ ] A bicycle pedal is pushed by 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. (Round your answer to one decimal place.) 10.6 N m 24. Question Details SEssCalc [ ] Find the magnitude of the torque about P if an F = 76lb force is applied as shown. (Round your answer to the nearest whole number.) 415 ftlb Page 13 of 19
14 25. Question Details SEssCalc [ ] (a) Find parametric equations for the line through (2, 3, 4) that is perpendicular to the plane x y + 2z = 6. (Use the parameter t.) (x(t), y(t), z(t)) = (b) In what points does this line intersect the coordinate planes? xyplane yzplane xzplane 26. Question Details SEssCalc [ ] Find a vector equation for the line segment from (3, 1, 4) to (7, 5, 3). (Use the parameter t.) r(t) = Page 14 of 19
15 27. Question Details SEssCalc [ ] Determine whether the lines L 1 and L 2 are parallel, skew, or intersecting. L 1 : x = 9 + 6t, y = 12 3t, z = 3 + 9t L 2 : x = 2 + 8s, y = 6 4s, z = s parallel skew intersecting If they intersect, find the point of intersection. (If an answer does not exist, enter DNE.) Since the direction vectors 6, 3, 9 and 8, 4, 10 are not scalar multiples of each other, the lines aren't parallel. For the lines to intersect, we must be able to find one value of t and one value of s that produce the same point from the respective parametric equations. Thus we need to satisfy the following three equations: 9 + 6t = 2 + 8s, 12 3t = 6 4s, 3 + 9t = s. Solving the last two equations we get t = 20, s = and checking, we see that these values don't satisfy the first equation. Thus the lines aren't parallel and don't intersect, so they must be skew lines. 28. Question Details SEssCalc [ ] Find an equation of the plane. The plane through the point (7, 1, 2) and parallel to the plane 9x y z = Question Details SEssCalc [ ] Find an equation of the plane. The plane through the points (0, 9, 9), (9, 0, 9), and (9, 9, 0) Page 15 of 19
16 30. Question Details SEssCalc [ ] Find an equation of the plane. The plane through the origin and the points (2, 2, 7) and (9, 2, 4) 31. Question Details SEssCalc [ ] Find an equation of the plane. The plane that passes through (9, 0, 2) and contains the line x = 7 2t, y = 1 + 3t, z = 3 + 2t 32. Question Details SEssCalc [ ] Find an equation of the plane. The plane that passes through the point x + y z = 5 and 4x y + 5z = 2 ( 2, 1, 2) and contains the line of intersection of the planes 33. Question Details SEssCalc [ ] Find an equation of the plane. The plane that passes through the line of intersection of the planes the plane x + y 4z = 3 x z = 2 and y + 4z = 1 and is perpendicular to Page 16 of 19
17 34. Question Details SEssCalc [ ] Find the point at which the line x = 5 t, y = 4 + t, z = 4t intersects the plane x y + 5z = Question Details SEssCalc [ ] Determine whether the planes are parallel, perpendicular, or neither. 9x + 9y + 9z = 1, 9x 9y + 9z = 1 parallel perpendicular neither If neither, find the angle between them. (Round your answer to one decimal place. If the planes are parallel or perpendicular, enter PARALLEL or PERPENDICULAR, respectively.) Normal vector for the planes are n 1 = 9, 9, 9 and n 2 = 9, 9, 9. The normals are not parallel, so neither are the planes. Furthermore, n 1 n 2 = = 81 0, so the planes aren't perpendicular. The angle between them is given by n cos θ = 1 n 2 81 = = 81 = 1 θ = cos n 1 n Question Details SEssCalc [ ] (a) Find parametric equations for the line of intersection of the planes x + y + z = 2 and x + 3y + 3z = 2. (x(t), y(t), z(t)) = (b) Find the angle between these planes. (Round your answer to one decimal place.) 22.0 Page 17 of 19
18 37. Question Details SEssCalc [ ] Find the distance from the point to the given plane. (1, 3, 8), 3x + 2y + 6z = 5 ax 1 + by 1 + cz 1 + d 3(1) + 2( 3) + 6(8) 5 40 By the equation D =, the distance is D = = = 40. a 2 + b 2 + c Question Details SEssCalc [ ] Find the distance from the point to the given plane. ( 9, 6, 5), x 2y 4z = 8 ax 1 + by 1 + cz 1 + d 1( 9) 2(6) 4(5) By the equation D =, the distance is D = = =. a 2 + b 2 + c ( 2) 2 + ( 4) Question Details SEssCalc [ ] Find the distance between the given parallel planes. 5x 5y + z = 20, 10x 10y + 2z = 2 Page 18 of 19
19 40. Question Details SEssCalc [ ] (a) Find the point at which the given lines intersect. r = 1, 2, 0 + t 2, 2, 2 r = 3, 0, 2 + s 2, 2, 0 (b) Find an equation of the plane that contains these lines. Assignment Details Name (AID): REVIEW FOR THE EXAM 1 MATH 203 F 2014 ( ) Submissions Allowed: 5 Category: Homework Code: Locked: No Author: Islam, Mohammad ( shafiqusa@gmail.com ) Last Saved: Sep 17, :56 AM EDT Permission: Protected Randomization: Person Which graded: Last Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Add Practice Button Help/Hints Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response Page 19 of 19
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