Scalar triple product (STP) is the combination of a dot product and a cross product, in the form of
p = a . (b x c)
where p is the STP,
a, b and c are (usually non-coplanar) vectors,
. is the dot product operator, and
x is the cross product operator.
The brackets are there to show the order of operation. The result is a scalar (hence the name).
^(c)
| ^(b)
| /
| /
| /
|/
+-------------->(a)
(To aid in understanding, assume b is pointing into or out of the page)
STP is especially useful for calculating the volume of any kind of sheared rectangular prism (eg a trapezoidal prism). It can also be used to test whether three vectors are coplanar.
If the three vectors are coplanar, the STP will be zero.
Proof:
If a, b and c are coplanar,
b x c will equal a vector perpendicular to a (1),
the dot product of perpendicular vectors equals zero (2).
(See the entries on dot product (for (2)) and cross product (for (1)) for the formulae that support these statements.