Triple Scalar Products

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

Suppose the vectors \textbf{u} = (u_{1}, u_{2}, u_{3}), \textbf{v} = (v_{1}, v_{2}, v_{3}) and \textbf{w} = (w_{1}, w_{2}, w_{3}) are in 3-space. The scalar triple product is the dot product of the vector \textbf{u} and the cross product vector of vectors \textbf{v} and \textbf{w}. Mathematically, the scalar triple product is represented as:

    \[\textbf{u} \cdot (\textbf{v} \times \textbf{w})\]

One could compute this triple scalar product by finding the cross product vector of \textbf{v} and \textbf{w} and then taking the dot product.

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

 

Source: http://quicklatex.com/cache3/3c/ql_724e6531f9c25f3e4b081c9951e7653c_l3.png


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.

 

Source: https://www.technologyuk.net/mathematics/geometry/images/geometry_0164.gif

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

    \[V = |\textbf{u} \cdot (\textbf{v} \times \textbf{w})|\]


An Important Result

Suppose we have the three vectors \textbf{u} = (u_{1}, u_{2}, u_{3}), \textbf{v} = (v_{1}, v_{2}, v_{3}) and \textbf{w} = (w_{1}, w_{2}, w_{3}). 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

 

Source: http://quicklatex.com/cache3/f4/ql_562949f79ef0aa60b12546877c7386f4_l3.png


Examples

Example One

Given the vectors \textbf{u} = (0, -1, 1), \textbf{v} = (2, 0, 3) and \textbf{w} = (-1, 3, 0), compute the scalar triple product \textbf{u} \cdot (\textbf{v} \times \textbf{w}).

Answer

One can show that the cross product \textbf{v} \times \textbf{w} is (-9, -3, 6). The triple scalar product computes as follows.

    \[\textbf{u} \cdot (\textbf{v} \times \textbf{w} = (0, -1, 1) \cdot (-9, -3, 6) = 0 + 3 + 6 = 9\]


Example Two

Compute the volume of the parallelpiped determined by the vectors \textbf{u} = (3, 0, -1), \textbf{v} = (1, 1, 1) and \textbf{w} = (0, 2, 0).

Answer

The cross product \textbf{v} \times \textbf{w} is (-2, 0, 2). The triple scalar product is evaluated as follows.

    \[\textbf{u} \cdot (\textbf{v} \times \textbf{w} = (3, 0, -1) \cdot (-2, 0, 2) = -6 + 0 - 2 = -8\]

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 \textbf{u} = (0, 0, 1), \textbf{v} = (1, -3, 4) and \textbf{w} = (1, 0, 2) 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.

Source: http://quicklatex.com/cache3/88/ql_33cba57fedeacaa263a2888254b45a88_l3.png

 

These three vectors do not lie in the same plane.


Practice Problems

1) Given the vectors \textbf{u} = (5, 0, 1), \textbf{v} = (2, 2, 3) and \textbf{w} = (0, 3, 0), compute the scalar triple product \textbf{u} \cdot (\textbf{v} \times \textbf{w}).

2) What is the volume of the parallelpiped determined by the vectors \textbf{u} = (0, 2, -1), \textbf{v} = (3, 0, 1) and \textbf{w} = (0, -1, 1)?

3) Suppose the vectors \textbf{u} = (5, -3, 2), \textbf{v} = (0, -2, 1) and \textbf{w} = (1, 0, -3). All three vectors have the same initial point. Are these vectors on the same plane?


Solutions

1) -39 (\textbf{v} \times \textbf{w} = (-9, 0, 6))

2) 3 cubic units. (\textbf{v} \times \textbf{w} = (1, -3, -3))

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 31 \neq 0. (\textbf{v} \times \textbf{w} = (6, 1, 2))


Reference

Elementary Linear Algebra (Tenth Edition) by Howard Anton