Find the volume of the parallelepiped formed by vectors $\mathbf{a}=(2,3,5)$, $\mathbf{b}=(-1,4,0)$ and $\mathbf{c}=(0,-2,3)$.

Step-by-step solution

1. Compute the cross product $\mathbf{b}\times \mathbf{c}$: \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\-1&4&0\0&-2&3\end{\vmatrix}=12\mathbf{i}+3\mathbf{j}+2\mathbf{k}.
2. Take the dot product of $\mathbf{a}$ with this result: $2(12)+3(3)+5(2)=24+9+10=43$.
3. The scalar triple product $|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$ gives the volume.
4. Thus the volume of the parallelepiped is $43$ cubic units.

Concept tested: Scalar triple product (volume of parallelepiped)

Published 10 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked