2.4 The cross product  (Page 5/16)

The triple scalar product

Because the cross product of two vectors is a vector, it is possible to combine the dot product and the cross product. The dot product of a vector with the cross product of two other vectors is called the triple scalar product because the result is a scalar.

Proof

The calculation is straightforward.

□

When we create a matrix from three vectors, we must be careful about the order in which we list the vectors. If we list them in a matrix in one order and then rearrange the rows, the absolute value of the determinant remains unchanged. However, each time two rows switch places, the determinant changes sign:

Verifying this fact is straightforward, but rather messy. Let’s take a look at this with an example:

Switching the top two rows we have

Rearranging vectors in the triple products is equivalent to reordering the rows in the matrix of the determinant. Let $\text{u}=u_1\text{i}+u_2\text{j}+u_3\text{k},$ $\text{v}=v_1\text{i}+v_2\text{j}+v_3\text{k},$ and $\text{w}=w_1\text{i}+w_2\text{j}+w_3\text{k}.$ Applying [link] , we have

We can obtain the determinant for calculating $\text{u}·\left(\text{w}\times \text{v}\right)$ by switching the bottom two rows of $\text{u}·\left(\text{v}\times \text{w}\right).$ Therefore, $\text{u}·\left(\text{v}\times \text{w}\right)=\text{−}\text{u}·\left(\text{w}\times \text{v}\right).$

Following this reasoning and exploring the different ways we can interchange variables in the triple scalar product lead to the following identities:

Let $\text{u}$ and $\text{v}$ be two vectors in standard position. If $\text{u}$ and $\text{v}$ are not scalar multiples of each other, then these vectors form adjacent sides of a parallelogram. We saw in [link] that the area of this parallelogram is $\Vert \text{u}\times \text{v}\Vert .$ Now suppose we add a third vector $\text{w}$ that does not lie in the same plane as $\text{u}$ and $\text{v}$ but still shares the same initial point. Then these vectors form three edges of a parallelepiped    , a three-dimensional prism with six faces that are each parallelograms, as shown in [link] . The volume of this prism is the product of the figure’s height and the area of its base. The triple scalar product of $\text{u},\text{v},$ and $\text{w}$ provides a simple method for calculating the volume of the parallelepiped defined by these vectors.