11:49 AM ... 1.1KB/s c 48% 286 4 GEOMETRICAL ASPECTS OF STANDARD SPACES Items (1)-(7) can be verified directly from the definition of cross product and properties of the dot product, while item (8) follows from (7), cquation (4.2) and the definition of dot product. Note that (8) has an interesting geometrical interpretation for vectors u, v E Rl, namely that u x v is the area of the parallelogram with adjacent sides represented by the vectors u and y. 4.1 Exercises and Problems Exercise 1. For the following pairs of vectors, calculate uv, ull, and ||vl. (a) (3,-5), (2,4) (b) (1.1,2), (2, -1,3) (c) (2, 1.-2.-1). (3.0.1,-4) (1+2i, 2+i), (1+3i, 1) (e) (3,1,2.-1), (2,0,1,1) (1) (2,2,-2), (2, 1,5) Exercise 2. For the following pairs of vectors, calculate u v and unit vectors in the direction of u and v. (a) (4, -2,2), (1, 3.2) (b) (1.1), (2,-2) (C) (4,0,1,2 - 3i), (1,1 - 21,1,i) (d) (i. -i), (31.1) (e) (1,-1,1.-1), (2.2.1,1) (f) (4,1,2), (1,0,0) Exercise 3. Let be the angle between the following pairs of real vectors and compute cose using dot products, (a) (2,-5), (4,2) (b) (3,4), (4,-3) (1,1,2), (2,-1,3) (d)j + k, 2i+k @4.1.2, (2. Exercise 4. Compute an angle between the following pairs of real vectors. (a) (4,5), (-4,4) (b) i - 5j, i+k (c) (4,0,2), (1,1,1) Exercise 5. Compute the cross product of the vector pairs in Exercise 4. (Ex- press two-dimensional vectors in terms of i and j first.) Exercise 6. Compute sin 0, where 0 is the angle between the following pairs of real vectors, using cross products. (a) 3i - 5j, 2i+ 4 (b) 3i - 5j + 2k, 2i - 4k (c) (-4, 2, 4), (4,1,-5) Exercise 7 Let c = 3, u =(4,-1.2.3), and v = (-2.2.-2.2). Verify that the four basic norm laws hold for these vectors and scalars. Exercise 8. Let c= 2. u= (-3.2.1), v = (4.2, -3), and w = (1.-2.1). Verify the four basic inner product laws for these vectors and scalars. Exercise 9. Let c= -2, u = (0,2.1), v = (4.0,-3), and w = (1, -2,1). Verify cross product laws (1)-(4) for these vectors and scalars. Exercise 10. Let u = (1.2.2), v = (0,2,-3), and w = (1,0,1). Verify cross product laws (5)-(7) for these vectors and scalars