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8-5 Dot and Cross Products of Vectors in Space 23. WEIGHTLIFTING A weightlifter doing bicep curls applies 212 newtons of force to lift the dumbbell. The weightlifter’s forearm is 0.356 meters long and she begins the bicep curl with her elbow bent at a 15º angle below the horizontal, in the direction of the positive x-axis. a. Find the vector representing the torque about the weightlifter’s elbow in component form. b. Find the magnitude and direction of the torque. SOLUTION: a. The component form of the vector representing the directed distance from the axis of rotation to the end of the weightlifter’s forearm can be found using the triangle shown and trigonometric ratios. Vector r is therefore or about . Notice that the z-component is negative since the weightlifter’s arm is at a 15° angle below the horizontal. The vector representing the force applied to the weight is 212 newtons up, so . Use the cross product of these vectors to find the vector representing the torque about the hinge. The component form of the torque vector is . b. The component form of the torque vector newton-meters parallel to the negative y-axis. tells us that the magnitude of the vector is about 72.08 Find the area of the parallelogram with adjacent sides u and v. 25. SOLUTION: First, find . eSolutions Manual - Powered by Cognero Page 1 The component form of the torque vector is . The component form of the torque vector tells us that the magnitude of the vector is about 72.08 8-5 b. Dot and Cross Products of Vectors in Space newton-meters parallel to the negative y-axis. Find the area of the parallelogram with adjacent sides u and v. 25. SOLUTION: First, find . Then, find the magnitude of . The area of the parallelogram is about 56.7 square units. 27. u = 6i – 2j + 5k , v = 5i – 4j – 8k SOLUTION: First, write u and v in component form as Then, find the magnitude of Next, find . . eSolutions Manual - Powered by Cognero Page 2 8-5 The Dotarea and Cross Products of Vectors Space of the parallelogram is about 56.7 squarein units. 27. u = 6i – 2j + 5k , v = 5i – 4j – 8k SOLUTION: First, write u and v in component form as Then, find the magnitude of Next, find . Next, find . . The area of the parallelogram is about 82.6 square units. 29. u = −3i – 5j + 3k , v = 4i – j + 6k SOLUTION: First, write u and v in component form as Then, find the magnitude of . eSolutions by Cognero TheManual area of- Powered the parallelogram is about 46.5 square units. Find the volume of the parallelepiped having t, u, and v as adjacent edges. Page 3 8-5 Dot and Cross Products of Vectors in Space The area of the parallelogram is about 82.6 square units. 29. u = −3i – 5j + 3k , v = 4i – j + 6k SOLUTION: First, write u and v in component form as Then, find the magnitude of Next, find . . The area of the parallelogram is about 46.5 square units. Find the volume of the parallelepiped having t, u, and v as adjacent edges. 31. SOLUTION: The volume of the parallelepiped is 206 cubic units. 33. t = −4i + j + 3k , u = 5i + 7j – 6k , v = 3i – 2j – 5k SOLUTION: Write t, u, and v in component form as eSolutions Manual - Powered by Cognero Page 4 8-5 The Dotvolume and Cross Productsisof in Space of the parallelepiped 206Vectors cubic units. 33. t = −4i + j + 3k , u = 5i + 7j – 6k , v = 3i – 2j – 5k SOLUTION: Write t, u, and v in component form as The volume of the parallelepiped is 102 cubic units. 35. t = 5i – 2j + 6k , u = 3i – 5j + 7k , v = 8i – j + 4k SOLUTION: Write t, u, and v in component form as The volume of the parallelepiped is 69 cubic units. Find a vector that is orthogonal to each vector. 37. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let and . Find the dot product of a and b. If a and b are orthogonal, then −x − 2y + 5z = 0. Substitute values for x and y and solve for z. Let x = 5 and y = 5. eSolutions Manual - Powered by Cognero A vector orthogonal to Page 5 is . 8-5 Dot and Cross Products of Vectors in Space The volume of the parallelepiped is 69 cubic units. Find a vector that is orthogonal to each vector. 37. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let and . Find the dot product of a and b. If a and b are orthogonal, then −x − 2y + 5z = 0. Substitute values for x and y and solve for z. Let x = 5 and y = 5. A vector orthogonal to is . 39. SOLUTION: Sample answer: Two vectors are orthogonal if and only if their dot product is equal to 0. Let and . Find the dot product of a and b. If a and b are orthogonal, then 7x + 8z = 0. Substitute a value for x and solve for z. Let x = −8. A vector orthogonal to is . Given v and u · v, find u. 41. v = , u · v = SOLUTION: Sample answer: Let eSolutions Manual - Powered by Cognero . Substitute u, v , and into the equation for a dot product. Page 6 8-5 A Dot and Cross Products of Vectors in Space vector orthogonal to . is Given v and u · v, find u. 41. v = , u · v = SOLUTION: . Substitute u, v , and Sample answer: Let into the equation for a dot product. Substitute a value for x and solve for z. Let x = −1. Therefore, . Determine whether the points are collinear. 43. SOLUTION: Let a = (−1, 7, 7), b = (−3, 9, 11), and c = (−5, 11, 13). Form two vectors, collinear, then the angle between and will be 0° or 180°. Find the component form of each vector. and . If the three points are Use the formula for the angle between two vectors. eSolutions Manual - Powered by Cognero Page 7 . 8-5 Therefore, Dot and Cross Products of Vectors in Space Determine whether the points are collinear. 43. SOLUTION: Let a = (−1, 7, 7), b = (−3, 9, 11), and c = (−5, 11, 13). Form two vectors, collinear, then the angle between and will be 0° or 180°. Find the component form of each vector. and . If the three points are Use the formula for the angle between two vectors. The angle between the vectors is not 180° nor 0°. Therefore, the points are not collinear. Determine whether each pair of vectors are parallel. 45. SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors. eSolutions Manual - Powered by Cognero Page 8 8-5 The angle between the vectors is not 180° nor 0°. Therefore, the points are not collinear. Dot and Cross Products of Vectors in Space Determine whether each pair of vectors are parallel. 45. SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors. The angle between the vectors is 0°. Therefore, the vectors are parallel. 47. w = , SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors. eSolutions Manual - Powered by Cognero Page 9 8-5 The angle between the vectors is 0°. Therefore, the vectors are parallel. Dot and Cross Products of Vectors in Space 47. w = , SOLUTION: If two vectors are parallel, then the angle between the vectors will be 0 or 180°. Use the formula for the angle between two vectors. The angle between the vectors is 0°. Therefore, the vectors are parallel. eSolutions Manual - Powered by Cognero Page 10