Question

Find the volumes of the tetrahedron with vertices $(-1,0,1),(2,-1,0),(3,2,5),(1,2,1)$

• A. $8$
• B. $4$
• C. $7$
• D. $6$

Answer 1

The correct option is D $6$

Let $A(-1,0,1),B(2,-1,0),C(3,2,5)$ and $D(1,2,1)$
$\overrightarrow{AB}=B-A=(2,-1,0)-(-1,0,1)=(3,-1,-1)\overrightarrow{AC}=C-A=(3,2,5)-(-1,0,1)=(4,2,4)\overrightarrow{AD}=(1,2,1)-(-1,0,1)=(2,2,0)$
$V=\frac{1}{6}\left|\begin{array}{ccc}3& -1& -1\\ 4& 2& 4\\ 2& 2& 0\end{array}\right|=\frac{1}{6}(3(8-0)+1(8-0)-1(4-8))$
$=\frac{1}{6}(24+8+4)=\frac{36}{6}=6$