Question

The volume of the tetrahedron with vertices $P(-1,2,0),Q(2,1,-3),R(1,0,1)$ and $S(3,-2,3)$ is

• A. $1/3$
• B. $2/3$
• C. $1$
• D. $4/5$

Answer 1

Answer: (B)

$2/3$

Given vertices of the tetrahedron are $P(-1,2,0),Q(2,1,-3),R(1,0,1)$ and $S(3,-2,3)$
The volume of a tetrahedron is equal to $\frac{1}{6}$ of the absolute value of the triple product.
$V=\frac{1}{6}\left[\overrightarrow{PQ}\overrightarrow{PR}\overrightarrow{PS}\right]$
$\to \overrightarrow{PQ}=3i-j-3k;\overrightarrow{PR}=2i-2j+kand\overrightarrow{PS}=4i-4j+3k$
$\Rightarrow V=\frac{1}{6}\left|\begin{array}{ccc}3& -1& -3\\ 2& -2& 1\\ 4& -4& 3\end{array}\right|$
$\therefore V=\frac{2}{3}$