The points $O(0,0,0)$, $A(2,1,3)$, $B(1,4,2)$ and $C(3,2,5)$ are the vertices of a tetrahedron. Find its volume. (Give your answer in cubic units.)

Step-by-step solution

1. Form the position vectors of the three edges meeting at $O$: $\mathbf{OA}=(2,1,3)$, $\mathbf{OB}=(1,4,2)$, $\mathbf{OC}=(3,2,5)$.
2. Compute the cross product $\mathbf{OB}\times \mathbf{OC}$: $\mathbf{OB}\times \mathbf{OC}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 4& 2\\ 3& 2& 5\end{array}\right|=(16,1,-10)$.
3. Take the dot product of $\mathbf{OA}$ with the above cross product: $\mathbf{OA}\cdot (\mathbf{OB}\times \mathbf{OC})=(2,1,3)\cdot (16,1,-10)=2\cdot 16+1\cdot 1+3\cdot (-10)=32+1-30=3$.
4. The absolute value of the scalar triple product gives $|\mathbf{OA}\cdot (\mathbf{OB}\times \mathbf{OC})|=3$.
5. Volume of a tetrahedron $=\frac{1}{6}$ times the scalar triple product, so $V=\frac{1}{6}\times 3=0.5$.
6. Thus the volume is $0.5$ cubic units.

Concept tested: Scalar triple product and volume of a tetrahedron

Published 8 September 2026 · How this solution was prepared and checked