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Consider a tetrahedron with vertices
. Prove that the volume of the tetrahedron is given by the formula
- Compute the volume in the case that
- Proof. We know the volume of a tetrahedron is given by
(where
denotes the altitude of the tetrahedron). We know (page 490 of Apostol) that the volume of the parallelepiped with base formed by vector
and height formed by vector
is given by
. In this case we have that the base of the tetrahedron is formed by the vectors
and
, and the height is formed by the vector
. Further, we know that the area of the base described by the vectors
and
is one half that of the parallelepiped whose base is given by vectors
and
(since the base of the parallelepiped described by vectors
and
is a rectangle, and the base of the tetrahedron is the triangle formed by cutting this rectangle along the diagonal). Therefore we have
- Using the formula in part (a) with the given values of
we have
Shouldn’t it be h denotes the altitude?