*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.