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

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Solution

$- - \to O A = \to a =^i + 2^j + 3^k, - - \to O B = \to b = 2^i + 3^j + 4^k$
$- - \to O C = \to C = 2^i + 3^j -^k$
Vol. of tetrahedron $= \frac{1}{6} [\to a \to b \to c]$
$= \frac{1}{6} ∣ ∣ ∣ (\to a \times \to b) . \to c ∣ ∣ ∣$
$= \frac{1}{6} ∣ ∣^i^j^k 123 234 ∣ ∣ . (2^i + 3^j -^k)$
Vol. $= \frac{1}{6} (^i (- 1) -^j (- 2) +^k (- 1)) . (2^i + 3^j -^k)$
$= \frac{1}{6} (- 2 + 6 + 1) = \frac{5}{6}$

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