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# Prove a vector formula for the volume of a tetrahedron

1. Consider a tetrahedron with vertices . Prove that the volume of the tetrahedron is given by the formula 2. Compute the volume in the case that 1. 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 2. Using the formula in part (a) with the given values of we have 