Question

Find the volume of the parallelepiped determined by the vectors a, b, and c.

a vector = (1,5,3) b vector = (-1,1,5) c vector = (5,1,2)

Im not sure if I have to a x (bxc) or the other way around. Please help and answer in cubic units.

Answer 1

The volume is given by any of the three permutations of the scalar triple product:

$|\mathbf a\cdot(\mathbf b\times\mathbf c)|=|\mathbf b\cdot(\mathbf c\times\mathbf a)|=|\mathbf c\cdot(\mathbf a\times\mathbf b)|$

In fact, the order in which you choose $\mathbf a,\mathbf b,\mathbf c$ doesn't matter. Although the cross product is anticommutative, i.e. $\mathbf a\times\mathbf b=-(\mathbf b\times\mathbf a)$, you end up taking the absolute value anyway, so you'll always have a positive number.

Also, $\mathbf a\cdot(\mathbf b\times \mathbf c)=(\mathbf a\times\mathbf b)\cdot\mathbf c$, so really it doesn't matter what order you use, so remembering how to find the volume is easy as long as you know to use a dot and cross product (not two cross products).

You should get a volume of 114 cubic units.