This page is about the triple scalar product. The triple scalar product combines determinants, dot products and cross products of vectors.

**Table Of Contents**

- The Triple Scalar Product
- Volume Of The Parallelepiped 3D Object
- An Important Result
- Examples
- Practice Problems
- Solutions
- Reference

**The Triple Scalar Product**

Suppose the vectors , and are in 3-space. The scalar triple product is the dot product of the vector and the cross product vector of vectors and . Mathematically, the scalar triple product is represented as:

One could compute this triple scalar product by finding the cross product vector of and and then taking the dot product.

Another way to compute this scalar triple product is computing the three by three determinant shown below.

**Volume Of The Parallelepiped 3D Object**

The rectangular prism is a fairly popular 3D object but what if the sides were paraellograms instead of rectangles? A three-dimensional object with the four sides as parallelograms and the top and bottom sides as rectangles is called a parallelpiped.

Here is a visual of the parallelpiped.

The volume of the parallelpiped is the absolute (positive) value of the triple scalar product.

**An Important Result**

Suppose we have the three vectors , and . If these vectors have the same initial point (i.e. The vectors start at the origin (0, 0, 0)) then they lie on the same plane if and only if

__Examples__

__Example One__

Given the vectors , and , compute the scalar triple product .

__Answer__

One can show that the cross product is (-9, -3, 6). The triple scalar product computes as follows.

__Example Two__

Compute the volume of the parallelpiped determined by the vectors , and .

__Answer__

The cross product is . The triple scalar product is evaluated as follows.

The volume of the parallelpiped is the absolute value of the scalar triple product. Taking the positive part of -8 would be 8. The parallelpiped’s volume is 8 cubic units.

__Example Three__

Suppose we have the vectors , and which have the same initial point. Compute the scalar triple product of the vectors. Do these vectors lie in the same plane?

__Answer__

We compute the three by three determinant (using cofactor expansion along the first row) here.

These three vectors do not lie in the same plane.

**Practice Problems**

1) Given the vectors , and , compute the scalar triple product .

2) What is the volume of the parallelpiped determined by the vectors , and ?

3) Suppose the vectors , and . All three vectors have the same initial point. Are these vectors on the same plane?

**Solutions**

1) -39 ()

2) 3 cubic units. ()

3) No. These three vectors are not on the same plane since the determinant of the 3 by 3 matrix (or the scalar triple product) is . ()

__Reference__

Elementary Linear Algebra (Tenth Edition) by Howard Anton