Why is the volume of a parallelepiped given three vectors zero?

Question

I'm trying to find the volume of a parallelepiped given three vectors:

• $A=(-2,3,1)$
• $B=(1,1,2)$
• $C=(1,2,3)$

So,

\begin{align*} V &= \| C \cdot (A \times B)\| \\ &= \| (1,2,3) \cdot \big((-2,3,1)\times (1,2,3)\big)\| \\ &= \| (1,2,3) \cdot (5,5,-5)\| \\ &= 0 \\ \end{align*}

Why is $0$? I checked the cross and dot products on Symbolab and they're fine. But I don't know what I'm missing. Any help?

Answer 1

Not a surprise since $A+7B-5C=0$ which means $A$,$B$ and $C$ are in the same plane.