Let , and . Then the scalar triple product is given by the formula

.

The proof is straightforward. Carrying out the scalar product of the vectors

and

we obtain

Geometric Interpretation: The absolute value of the number is the volume of the parallelepiped constructed on the vectors a, b and c as it is shown in the figure below:

Proof: The volume of a parallelepiped is equal to the product of the area of the base and its height.

By the theorem of scalar product,

,

where the quantity equals the area of the parallelogram, and the product equals the height of the parallelepiped.

Hence, the theorem.