Answer to Question #178942 in Analytic Geometry for John withers

Question

A parallelepiped is defined by the following vectors |𝑎| = (2, 3, −1), |𝑏| = (−1, 0, 2),

and |𝑐| = (3, −1, 2). Find the volume of the parallelepiped. 

Answer 1

The volume is given by the formula $|a\cdot |b\times c||.$ We can write $b= -\hat{i}+2\hat{k}$ and $c=3\hat{i}-\hat{j}+2\hat{k}.$ $b\times c=\hat{i}(-2\times -1)+\hat{j}(2\times 3-(-1)\times 2)+\hat{k}(-1\times -1)$ $=2\hat{i}+8\hat{j}+\hat{k}.$ Hence $a\cdot(b\times c)=$ $(2\hat{i}+3\hat{j}-\hat{k})\cdot (2\hat{i}+8\hat{j}+\hat{k})$ = $2\times(2)+(3)\times 8-(1\times 1)=27.$ Hence volume =$|27|=27$ in cube units.