5. Cross Product

b. Cross Product and Triple Product

2. Algebraic Definition of Triple Products

Before we can give the geometrical description of the cross product, we need to understand the triple product which is a combination of the dot product and the cross product.

The triple product of three vectors \(\vec u\), \(\vec v\) and \(\vec w\) is the scalar \(\vec u\cdot\vec v\times\vec w\).

We do not need parentheses in the formula \(\vec u\cdot\vec v\times\vec w\) because the order \((\vec u\cdot\vec v)\times\vec w\) would not make any sense; we cannot take the cross product of a scalar and a vector. So \(\vec u\cdot\vec v\times\vec w=\vec u\cdot(\vec v\times\vec w)\).

We want a component/determinant formula for the triple product.

Let \(\vec u=\left\langle u_1,u_2,u_3\right\rangle\), \(\vec v=\left\langle v_1,v_2,v_3\right\rangle\) and \(\vec w=\left\langle w_1,w_2,w_3\right\rangle\). Then \[ \vec v\times\vec w =\left\langle v_2w_3-v_3w_2,v_3w_1-v_1w_3,v_1w_2-v_2w_1\right\rangle \] and \[ \vec u\cdot\vec v\times\vec w =u_1(v_2w_3-v_3w_2) +u_2(v_3w_1-v_1w_3) +u_3(v_1w_2-v_2w_1) \] On the other hand, consider the determinant whose rows are \(\vec u\), \(\vec v\) and \(\vec w\), and expand on the first row: \[\begin{aligned} \det \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix} &=u_1 \begin{vmatrix} v_2 & v_3 \\ w_2 & w_3 \end{vmatrix} -u_2 \begin{vmatrix} v_1 & v_3 \\ w_1 & w_3 \end{vmatrix} +u_3 \begin{vmatrix} v_1 & v_2 \\ w_1 & w_2 \end{vmatrix} \\ &=u_1(v_2w_3-v_3w_2) -u_2(v_1w_3-v_3w_1) +u_3(v_1w_2-v_2w_1) \\ &=u_1(v_2w_3-v_3w_2) +u_2(v_3w_1-v_1w_3) +u_3(v_1w_2-v_2w_1) \end{aligned}\] This is the same as the triple product. So we have:

The triple product of \(\vec u=\left\langle u_1,u_2,u_3\right\rangle\), \(\vec v=\left\langle v_1,v_2,v_3\right\rangle\) and \(\vec w=\left\langle w_1,w_2,w_3\right\rangle\) may be computed using the determinant: \[ \vec u\cdot\vec v\times\vec w =\begin{vmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix} \] which has \(\vec u\) on the first row, \(\vec v\) on the second row and \(\vec w\) on the third row.

Properties

Since a determinant changes sign when two rows are interchanged, the triple product changes sign when any two vectors are interchanged. This combined with the commutativity of the dot product gives \(12\) formulas for the triple product:

Thus you get the same triple product no matter which order you enter the vectors, except for a possible minus sign. Two of these reorderings are especially notable:

Finally, since a determinant is zero when two rows are equal, the triple product is zero when any two vectors are equal.

Before we can give geometrical descriptions of the cross product and of the triple product, we need to know more about the properties of the cross product.