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. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{4}$
• D. $\frac{3}{4}$

Answer 1

Answer: (B)

Given : The vertices of tetrahedron are
P(-1, 2, 0), Q(2, 1, -3), R(1, 0, 1) & S(3, -2, 3)
$\therefore$ Volume of tetrahedron $=\frac{1}{6}[\overrightarrow{PQ}\overrightarrow{PR}\overrightarrow{PS}]$
Now,
$\overrightarrow{PQ}=(2+1)\hat{i}+(1-2)\hat{j}+(-3)\hat{k}=3\hat{i}-\hat{j}-3\hat{k}$
Similarly, $\overrightarrow{PR}=2\hat{i}-2\hat{j}+\hat{k}$
$\&\overrightarrow{PS}=4\hat{i}-4\hat{j}+3\hat{k}$
$\therefore$ Volume of tetrahedron
$=\frac{1}{6}\left|\begin{array}{ccc}3& -1& -3\\ 2& -2& 1\\ 4& -4& 3\end{array}\right|=\frac{2}{3}$