If a tetrahedron is
defined by three
vectors, the
volume of that tetrahedron can be found by
evaluating the
value of one-sixth of the
scalar triple product. So, if the tetrahedron has
corners at
points A, B, C and D, the three vectors needed are those in the
directions AB, AC and AD. Then the volume V is found by
V = (1/6)(AD.ABxAC).
As an example, let us consider a tetrahedron which is defined by the following vectors:
AB = 5i - j - k
AC = 2i - 8j + k
AD = -i + 2k
A quick way to evaluate the scalar triple product is to calculate the modulus of the determinant of a matrix consisting of these three vectors:
| / 5 -1 -1 \ |
V = (1/6)| det | 2 -8 1 | |
| \-1 0 2 / |
= (1/6)|5(-16 - 0) + 1(4 + 1) - 1(0 - 8)|
= (1/6)|-80 + 5 + 8|
= (1/6)|-67|
= (1/6)(67)
= 67/6
Therefore the tetrahedron has volume 67/6 units^3.