Find the volume of the tetrahedron whose vertices are $A (3, 2, 6) B (6, 7, 8), C (7, 11, 23)$ and $D (9, 12, 17)$

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Solution

Given $a = - - \to A B = 3 ˆ i + 5 ˆ j + 2 ˆ k b = - - \to A C = 4 ˆ i + 9 ˆ j + 17 ˆ k c = - - \to A D = 6 ˆ i + 10 ˆ j + 11 ˆ k$
volume of the tetrahedron $\frac{1}{6} | (a \times b) . c | = \frac{1}{6} ∣ ∣ ((3 ˆ i + 5 ˆ j + 2 ˆ k) \times (4 ˆ i + 9 ˆ j + 17 ˆ k)) . (6 ˆ i + 10 ˆ j 11 ˆ k) ∣ ∣ = \frac{1}{6} ∣ ∣ ((85 - 18) ˆ i + (8 - 51) ˆ j + (27 - 20) ˆ k) . (6 ˆ i + 10 ˆ j + 11 ˆ k) ∣ ∣ = \frac{1}{6} ∣ ∣ (67 ˆ i - 43 ˆ j + 7 ˆ k) . (6 ˆ i + 10 ˆ j + 11 ˆ k) ∣ ∣ = \frac{1}{6} | 402 - 430 + 77 | = \frac{1}{6} \times 49 = \frac{49}{6}$

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