Recall that the area of a parallelogram with base b and altitude h is given by bh, and the volume of a parallelepiped is given by the area of its base times its altitude.
a) Let a, b ∈ R3 be nonzero vectors and V represent the parallelogram
{(x, y, z) = ua + vb : u, v ∈ [0, 1]}.
Prove that the area of V is ||a x b||.
b) Let a, b, c ∈ R3 be nonzero vectors and V represent the parallelepiped
{(x, y, z) = ta + ub + vc : t, u, v ∈ [0, 1]}.
Prove that the volume of P is |(a × b) ∙ c|.