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

Step-by-step solution

1. Step 1: Recall that the volume is the absolute value of the scalar triple product: $V=|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|$.
2. Step 2: Compute the cross product $\mathbf{b}\times \mathbf{c}$.
3. Step 3: Set up the determinant: $\left|\begin{array}{ccccccc}\mathbf{i}& \mathbf{j}& \mathbf{k}4& 0& -12& -3& 5\end{array}\right|$.
4. Step 4: Expand: $\mathbf{i}(0\cdot 5-(-1)(-3))-\mathbf{j}(4\cdot 5-(-1)(2))+\mathbf{k}(4\cdot (-3)-0\cdot 2)$.
5. Step 5: Simplify each term: $\mathbf{i}(0-3)-\mathbf{j}(20-(-2))+\mathbf{k}(-12-0)=-3\mathbf{i}-22\mathbf{j}-12\mathbf{k}$.
6. Step 6: So $\mathbf{b}\times \mathbf{c}=(-3,-22,-12)$.
7. Step 7: Compute the dot product with $\mathbf{a}$: $\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=(1,2,3)\cdot (-3,-22,-12)$.
8. Step 8: Perform the dot product: $1\cdot (-3)+2\cdot (-22)+3\cdot (-12)=-3-44-36=-83$.
9. Step 9: Take the absolute value: $V=|-83|=83$.
10. Step 10: Conclude the volume of the parallelepiped is $83$ cubic units.

Concept tested: Triple scalar product

Published 18 September 2026 · How this solution was prepared and checked