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37. Use the scalar triple product to verify that the vectors u = i + 5j - 2k, v = 3i - j, and w = 5i + 9j - 4k are coplanar.

37. Use the scalar triple product to verify that the vectors u = i + 5j - 2k, v = 3i - j, and w = 5i + 9j - 4k are coplanar.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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822 32. P(2, -3, 4), Q(-1, -2, 2), R(3, 1, -3) CHAPTER 12 Vectors and the Geometry of Space 33-34 Find the volume of the parallelepiped determined by the vectors a, b, and c. 33. a = (1, 2, 3), b = (-1, 1, 2), c = (2, 1, 4) 34. a = i + j, b=j+k, c=i+j+k 250 35-36 Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 35. P(-2, 1, 0), Q(2, 3, 2), 36. P(3, 0, 1), Q(-1, 2, 5), 37. Use the scalar triple product to verify that the vectors u = i + 5j-2 k, v = 3i- j, and w = 5i + 9j - 4k are coplanar. The 38. Use the scalar triple product to determine whether the points A(1, 3, 2), B(3, -1, 6), C(5, 2, 0), and D(3, 6, -4) lie in the same plane. 39. A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magni- tude of the torque about P. 2 ft 0.6 ft R(1, 4, -1), R(5, 1, -1), 60 N P 0.6 ft 70° 40. (a) A horizontal force of 20 lb is applied to the handle of a S gearshift lever as shown. Find the magnitude of the torque about the pivot point P. (b) Find the magnitude of the torque about P if the same force is applied at the elbow Q of the lever. S(3, 6, 1) S(0, 4, 2) 1 ft 10⁰ Q P 20 lb 41. A wrench 30 cm long lies along the positive y-axis 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 supply 100 Nm of torque to the bolt. 42. Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find the maximum and minimum values of the length of the vector u X v. In what direction does u X v point? 43. If a b = √3 and a × b = (1, 2, 2), find the angle between a and b. 44. (a) Find all vectors v such that (1, 2, 1) × v = (3, 1,-5) EX X (b) Explain why there is no vector v such that (1, 2, 1) X v = (3, 1,5) 45. (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 Lis d= where a = QR and b = QP. (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 6, 8) and R(-1,4,7). 46. (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 = axb| a 52. Prove that |a (bx c) | axb| . where a = QR, b = Qs, and c = QP. (b) Use the formula in part (a) to find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3). 47. Show that a × b ² = |a|²|b|² - (a - b)². 48. If a + b + c = 0, show that aXb=bXc=cXa 49. Prove that (a - b) x (a + b) = 2(a X b). 50. Prove Property 6 of cross products, that is, ax (bx c) = (ac)b - (a - b)c 51. Use Exercise 50 to prove that ax (bx c) + bx (cx a) + cx (ax b) = 0 (a X b) (cx d) = a c b c a d b d
Transcribed Image Text:822 32. P(2, -3, 4), Q(-1, -2, 2), R(3, 1, -3) CHAPTER 12 Vectors and the Geometry of Space 33-34 Find the volume of the parallelepiped determined by the vectors a, b, and c. 33. a = (1, 2, 3), b = (-1, 1, 2), c = (2, 1, 4) 34. a = i + j, b=j+k, c=i+j+k 250 35-36 Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 35. P(-2, 1, 0), Q(2, 3, 2), 36. P(3, 0, 1), Q(-1, 2, 5), 37. Use the scalar triple product to verify that the vectors u = i + 5j-2 k, v = 3i- j, and w = 5i + 9j - 4k are coplanar. The 38. Use the scalar triple product to determine whether the points A(1, 3, 2), B(3, -1, 6), C(5, 2, 0), and D(3, 6, -4) lie in the same plane. 39. A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magni- tude of the torque about P. 2 ft 0.6 ft R(1, 4, -1), R(5, 1, -1), 60 N P 0.6 ft 70° 40. (a) A horizontal force of 20 lb is applied to the handle of a S gearshift lever as shown. Find the magnitude of the torque about the pivot point P. (b) Find the magnitude of the torque about P if the same force is applied at the elbow Q of the lever. S(3, 6, 1) S(0, 4, 2) 1 ft 10⁰ Q P 20 lb 41. A wrench 30 cm long lies along the positive y-axis 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 supply 100 Nm of torque to the bolt. 42. Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find the maximum and minimum values of the length of the vector u X v. In what direction does u X v point? 43. If a b = √3 and a × b = (1, 2, 2), find the angle between a and b. 44. (a) Find all vectors v such that (1, 2, 1) × v = (3, 1,-5) EX X (b) Explain why there is no vector v such that (1, 2, 1) X v = (3, 1,5) 45. (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 Lis d= where a = QR and b = QP. (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 6, 8) and R(-1,4,7). 46. (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 = axb| a 52. Prove that |a (bx c) | axb| . where a = QR, b = Qs, and c = QP. (b) Use the formula in part (a) to find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3). 47. Show that a × b ² = |a|²|b|² - (a - b)². 48. If a + b + c = 0, show that aXb=bXc=cXa 49. Prove that (a - b) x (a + b) = 2(a X b). 50. Prove Property 6 of cross products, that is, ax (bx c) = (ac)b - (a - b)c 51. Use Exercise 50 to prove that ax (bx c) + bx (cx a) + cx (ax b) = 0 (a X b) (cx d) = a c b c a d b d
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