The reason is because half the area of a parallelogram is the area of a triangle. Say you have k linearly independent vectors, the volume of the parallelepiped they span is given by the formula:V(x_1,...,x_k)= \sqrt{det(X^{tr}X)}. This formula is easiest to understand in 3 dimensions. Say you have two vectors a and b which are elements of \mathbb{R}^3 then the volume formula says that V(a,b)^2=det\begin{bmatrix}<br />
\|a\|^2 & \langle a,b \rangle \\<br />
\langle b,a \rangle & \|b\|^2 \\<br />
\end{bmatrix} = \|a\|^2\|b\|^2-\langle a,b \rangle^2 = \|a\|^2\|b\|^2(1-\cos^2(\theta)) = \|a\|^2\|b\|^2\sin^2(\theta) which is just the area of the parallelogram spanned by a and b. This is the well known formula that magnitude of the cross product of the two vectors is the area of the parallelogram spanned by the two vectors.
