The cross product of two vectors in R2 is defined as the scalar
V x w = v1w2 - v2w1 (3.22)
for v = (v1, v2)T, w = (w1, w2)T.
(a) Does the cross product define an inner product on R2? Carefully explain which axioms are valid and which are not.
(b) Prove that v x w = ||v|| ||w|| sin#, where 9 denotes the angle from v to w as in Figure 3.2.
(c) Prove that v x w = 0 if and only if v and w are parallel vectors.
(d) Show that |v x w| equals the area of the parallelogram defined by v and w.