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Volume of a a parallelepiped

Suppose three vectors $ {\bf v } , {\bf u}$ and $ {\bf w}$ in three dimensional space $ R^3 $ are given so that they do not lie in the same plane. These three vectors form three edges of a parallelepiped. The volume of this parallelepiped ( is the product of area of the base and altitude ) is equal to the scalar triple product ${\bf u \cdot ( v \times w) }$ .

It can be shown that the volume of the parallelepiped is the absolute value of the determinant of the following matrix:


$A=\left[ \begin{array}{rrr}
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
w_1 & w_2 & w_3 \\
\end {array}


Ali A. Daddel 2000-09-15