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

Step-by-step solution

1. Form the scalar triple product matrix: $\left|\begin{array}{ccccccc}1& 0& 2\text{0}& 3& 1\text{2}& 1& 0\end{array}\right|$.
2. Expand along the first row: $1\cdot \left|\begin{array}{ccc}3& 1\text{1}& 0\end{array}\right|-0\cdot \left|\begin{array}{ccc}0& 1\text{2}& 0\end{array}\right|+2\cdot \left|\begin{array}{ccc}0& 3\text{2}& 1\end{array}\right|$.
3. Compute the small determinants: $\left|\begin{array}{ccc}3& 1\text{1}& 0\end{array}\right|=3\cdot 0-1\cdot 1=-1$, $\left|\begin{array}{ccc}0& 3\text{2}& 1\end{array}\right|=0\cdot 1-3\cdot 2=-6$.
4. Plug in: $1(-1)+0+2(-6)=-1-12=-13$.
5. The volume is the absolute value: $|-13|=13$.
6. Hence the volume of the parallelepiped is $13$.

Concept tested: Scalar triple product and volume

Published 12 September 2026 · How this solution was prepared and checked