Find the volume of the tetrahedron $V - A B C$ if $V$ is $(3, 2, - 4), A (4, 3, 3), B (3, 2, 1)$ and $C (1, 2, - 1)$

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Solution

Given vertices are $: V (3, 2, - 4); A (4, 3, 3); B (3, 2, 1); C (1, 2, - 1)$
In vector form :
$\Rightarrow V = 3 i + 2 j - 4 k; A = 4 i + 3 j + 3 k; B = 3 i + 2 j + k; C = i + 2 j - k$
We know volume of tetrahedron -
$= \frac{1}{6} [| (A - V) . (B - V) \times (C - V) |]$

$= \frac{1}{6} [| (i + j + 7 k) . (5 k) \times (- 2 i + 3 k) |]$

$= \frac{1}{6} [| (i + j + 7 k) . (10 j) |]$

$= \frac{1}{6} \times 10$

$= \frac{10}{6} .$
Hence, the answer is $\frac{10}{6} .$

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