display | more...

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.

Log in or register to write something here or to contact authors.