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19. Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0). unotors orthogonal to both i- k and i + j.

19. Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0). unotors orthogonal to both i- k and i + j.

Physics for Scientists and Engineers
Physics for Scientists and Engineers
10th Edition
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter11: Angular Momentum
Section: Chapter Questions
Problem 4P: Use the definition of the vector product and the definitions of the unit vectors i, j, and k to...
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10. The 1 12.4 EXERCISES 1-7 Find the cross product a X b and verify that it is orthogonal to both a and b. = (2, 3, 0), b = (1, 0, 5) (2, 3, 0), 1. a = 2. a = (4, 3, -2), b = (2, -1, 1) 3. a = 2j- 4k, b = -i + 3j + k 4. a = 3i+3j - 3k, b=3i - 3j + 3k 5. a =i+j+k, b=i+2j-3k 6. a = ti + cos tj + sin tk, b = i sin tj + cos / k 7. a = (1, 1, 1/1), b = (t², t², 1) 8. If a=i- 2k and b = j + k, find a X b. Sketch a, b, and a X b as vectors starting at the origin. 9-12 Find the vector, not with determinants, but by using properties of cross products. 9. (ix j) × k 11. (j-k) x (k - i) (a) a (b x c) • 13. State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (c) ax (bx c) (e) (a - b) x (cd) 10. kx (i 2j) 12. (i + j) × (i − j) 45° |u| = 4 - (b) a x (b. c) (d) a (b c) (f) (axb) (cx d) 14-15 Find |ux v and determine whether u X v is directed into the page or out of the page. 14. |v|=5 15. |u|=12 ● |v|= 16 120° 16. The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a| = 3 and | b | = 2. (a) Find a × b. (b) Use the right-hand rule to decide whether the components of a X b are positive, negative, or 0. 17. If a = 18. If a XX = ZA b a (1, 0, 1), b = (2, 1, -1), and c = (0, 1, 3), show that ax (bx c) = (a X b) X c. (2, -1, 3) and b = (4, 2, 1), find a × b and b X a. y 19. Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0). 20. Find two unit vectors orthogonal to both j - k and i + j. 21. Show that 0 Xa=0= ax 0 for any vector a in V3. 22. Show that (a X b) b = 0 for all vectors a and b in V3. 23-26 Prove the property of cross products (Theorem 11). 23. Property 1: a × b = -b xa 24. Property 2: (ca) X b = c(a X b) = a × (cb) 25. Property 3: a X (b + c) = a × b + ax c 26. Property 4: (a + b) × c = ax c + bxc 27. Find the area of the parallelogram with vertices A(-3,0), B(-1,3), C(5, 2), and D(3, -1). 28. Find the area of the parallelogram with vertices P(1, 0, 2), Q(3, 3, 3), R(7, 5, 8), and S(5, 2, 7). Q(4, 1, -2), 29-32 (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. 29. P(1, 0, 1), 30. P(0, 0, -3), 31. P(0, -2, 0), Q(-2, 1, 3), Q(4, 2, 0), R(4, 2, 5) R(3, 3, 1) R(5, 3, 1)
Transcribed Image Text:10. The 1 12.4 EXERCISES 1-7 Find the cross product a X b and verify that it is orthogonal to both a and b. = (2, 3, 0), b = (1, 0, 5) (2, 3, 0), 1. a = 2. a = (4, 3, -2), b = (2, -1, 1) 3. a = 2j- 4k, b = -i + 3j + k 4. a = 3i+3j - 3k, b=3i - 3j + 3k 5. a =i+j+k, b=i+2j-3k 6. a = ti + cos tj + sin tk, b = i sin tj + cos / k 7. a = (1, 1, 1/1), b = (t², t², 1) 8. If a=i- 2k and b = j + k, find a X b. Sketch a, b, and a X b as vectors starting at the origin. 9-12 Find the vector, not with determinants, but by using properties of cross products. 9. (ix j) × k 11. (j-k) x (k - i) (a) a (b x c) • 13. State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (c) ax (bx c) (e) (a - b) x (cd) 10. kx (i 2j) 12. (i + j) × (i − j) 45° |u| = 4 - (b) a x (b. c) (d) a (b c) (f) (axb) (cx d) 14-15 Find |ux v and determine whether u X v is directed into the page or out of the page. 14. |v|=5 15. |u|=12 ● |v|= 16 120° 16. The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a| = 3 and | b | = 2. (a) Find a × b. (b) Use the right-hand rule to decide whether the components of a X b are positive, negative, or 0. 17. If a = 18. If a XX = ZA b a (1, 0, 1), b = (2, 1, -1), and c = (0, 1, 3), show that ax (bx c) = (a X b) X c. (2, -1, 3) and b = (4, 2, 1), find a × b and b X a. y 19. Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0). 20. Find two unit vectors orthogonal to both j - k and i + j. 21. Show that 0 Xa=0= ax 0 for any vector a in V3. 22. Show that (a X b) b = 0 for all vectors a and b in V3. 23-26 Prove the property of cross products (Theorem 11). 23. Property 1: a × b = -b xa 24. Property 2: (ca) X b = c(a X b) = a × (cb) 25. Property 3: a X (b + c) = a × b + ax c 26. Property 4: (a + b) × c = ax c + bxc 27. Find the area of the parallelogram with vertices A(-3,0), B(-1,3), C(5, 2), and D(3, -1). 28. Find the area of the parallelogram with vertices P(1, 0, 2), Q(3, 3, 3), R(7, 5, 8), and S(5, 2, 7). Q(4, 1, -2), 29-32 (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. 29. P(1, 0, 1), 30. P(0, 0, -3), 31. P(0, -2, 0), Q(-2, 1, 3), Q(4, 2, 0), R(4, 2, 5) R(3, 3, 1) R(5, 3, 1)
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