Question

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(2, -2, 9)


(-7, 0, -3)


(-1, 7, 4)


(8, -2, -5)

Answer 1

If these are the vertices of a tetrahedron, its volume is |det(B − A, C − A, D − A)|/6. Take A = (2,-2,9), B = (−7,0,−3), C = (−1,7,4), D = (8,−2,−5).

Compute the edge vectors from A:
B − A = (−9, 2, −12)
C − A = (−3, 9, −5)
D − A = (6, 0, −14)

Compute (C − A) × (D − A):
(−3,9,−5) × (6,0,−14) = (−126, −72, −54).

Scalar triple product:
(B − A) · [(C − A) × (D − A)] = (−9,2,−12)·(−126,−72,−54) = 1638.

Volume = |1638|/6 = 273.

So the tetrahedron volume is 273 cubic units.