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  4. 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

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Q. 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

VITEEEVITEEE 2006

Solution:

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 $= \ \frac{1}{6} \ [\vec{PQ} \ \vec{PR} \ \vec{PS}]$
Now,
$\vec{PQ}=(2+1)\hat{i}+(1-2)\hat{j}+(-3)\hat{k}=3\hat{i}-\hat{j}-3\hat{k}$
Similarly, $\vec{PR}=2\hat{i}-2\hat{j}+\hat{k}$
$\& \ \ \ \ \ \vec{PS}=4\hat{i}-4\hat{j}+3\hat{k}$
$\therefore $ Volume of tetrahedron
$=\frac{1}{6} \begin{vmatrix} 3 & -1 & -3\\ 2 & -2 & 1 \\ 4 & -4 & 3 \end{vmatrix} =\frac{2}{3}$