Question

The volume of the tetrahedron whose vertices are $(0,1,2),(4,3,6),(2,3,2)$ and $(3,0,1)$ is $($ in unit${}^3)$

• A. $0$
• B. $1$
• C. $6$
• D. $3$

Answer 1

The correct option is C $6$

The volume of tetrahedron is given by $\frac{1}{6}[\overrightarrow{a}-\overrightarrow{d}\overrightarrow{b}-\overrightarrow{d}\overrightarrow{c}-\overrightarrow{d}]$ where $A=(0,1,2),B=(4,3,6),C=(2,3,2),D=(3,0,1)$

i.e. $\frac{1}{6}\left|\begin{array}{ccc}-3& 1& 1\\ 1& 3& 5\\ -1& 3& 1\end{array}\right|$

$=\frac{1}{6}\left[\right(-3\left)\right(3-15)-1(1+5)+1(3+3\left)\right]=6-1+1=6$