(a) Let a and b be vectors in R^3 . Express the area of the triangle formed by a, b, and a b in terms of a b. [1 point]

(b) Let P1,P2 P6 be a convex hexagon in R^2 with the property that opposites sides are parallel. Furthermore, suppose that the point P1 is at the origin (0, 0). On a piece of paper, draw this figure making sure to label the points in correct order. [1 point]

(c) Using the cross product, write down three equations that represent the fact that opposite sides of P1P2P3P4P5P6 are parallel. [3 points]

(d) Use the fact that P1 = 0 and properties of the cross product to show that your equations in part (c) may be simplified to

P2 P5 = P2 P4 (1)

P3 P6 = P4 P6 (2)

P3 P5 = P3 P6 P2 P6 + P2 P5. (3) [1 point]

(e) Use the equations in part (d) to show that the triangle P1P3P5 and triangle P2P4P6 have equal area. [4 points]