display | more...
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.