ni- 49. Use the cross product to find the area of the triangle in the xy-plane WO defined by (1, 2), (3, 4), and (-2, 2). X () nolupo ads pulos le- 50. Use the cross product to find the area of the quadrilateral in the xy- plane defined by (0, 0), (1, -1), (3, 1), and (2, 4). 51. Check that the four points P(2, 4, 4), Q(3, 1, 6), R(2, 8, 0), and S(7, 2, 1) all lie in a plane. Then use vectors to find the area of the quadri- lateral they define. 52. Use the cross product to find the area of the triangle with vertices (a, 0, 0), (0, b, 0), and (0, 0, c). In Exercises 53-55, prove each of the identities using the formula for the cross product. 53. V X W =-WxV 54. (2v) x w= 2(v x w) (2 a scalar) older 55. (utv) x w =uxw+vxw 56. Use the geometric description in Theorem 1 to prove Theorem 2 (iii): v x w =0 if and only if w = Av for some scalar A or v = 0. 57. Prove i x j = k and k x j = -i by each of the following methods: (a) Using the definition of cross product as a determinant (b) Using the geometric description of the cross product in Theorem 1 58. Using standard basis vectors, find an example demonstrating that the cross product is not associative.77. Let a, b, c be nonzero vectors. Assume that b and c are not parallel, and set v =a x (b xc), w = (a . c)b - (a . b)c (a) Prove that: (i) v lies in the plane spanned by b and c. (ii) v is orthogonal to a. (b) Prove that w also satisfies (i) and (ii). Conclude that v and w are par- allel. (c) Show algebraically that v = w (Figure 23). b xc and boy famod a a x (b x c) b FIGURE 2381. Assume that v and w lie in the first quadrant in R2 as in Figure 24. Use geometry to prove that the area of the parallelogram is equal to det warD'mol islogib C a (a + c, b + d) b (c, d) d W d (a, b) V a FIGURE 24