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Compute the volume of a tetrahedron with given vertices

by RoRi

Consider the tetrahedron with vertices at the origin and at the points where the plane

    \[ x + 2y + 3z = 6 \]

intersects the coordinate axes. Compute the volume of this tetrahedron.

First, the intercepts of the plane are given by (6,0,0), (0,3,0), (0,0,2). Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices A,B,C,D is

    \[ V = \frac{1}{6} | (B-A) \cdot (C -A) \times (D-A) |. \]

Letting A = (0,0,0), \ B = (6,0,0), \ C = (0,3,0), \ D = (0,0,2) we have

    \[ V = \frac{1}{6} | (6,0,0) \cdot (0,3,0) \times (0,0,2) | = \frac{1}{6} | (6,0,0) \cdot (6,0,0) | = 6. \]

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