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

Step-by-step solution

1. Compute the cross product $\mathbf{b}\times \mathbf{c}$.
2. $\begin{array}{rl}\mathbf{b}\times \mathbf{c}& =\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 4& -1& 2\\ 0& 5& -1\end{array}\right|\\ & =\mathbf{i}[(-1)(-1)-2\cdot 5]-\mathbf{j}[4(-1)-2\cdot 0]+\mathbf{k}[4\cdot 5-(-1)\cdot 0]\\ & =\mathbf{i}(1-10)-\mathbf{j}(-4)+\mathbf{k}(20)\\ & =(-9,4,20).\end{array}$
3. Take the scalar triple product $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$:
4. $\mathbf{a}\cdot (-9,4,20)=1\times (-9)+2\times 4+3\times 20=-9+8+60=59$.
5. The volume is the absolute value: $|59|=59$ cubic units.

Concept tested: Scalar triple product (volume of parallelepiped)

Published 5 September 2026 · How this solution was prepared and checked