The scalar triple product

Figure 10:

Consider three vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$. The scalar triple product is defined ${\bf a}\cdot {\bf b}\times {\bf c}$. Now, ${\bf b}\times {\bf c}$ is the vector area of the parallelogram defined by ${\bf b}$ and ${\bf c}$. So, ${\bf a}\cdot {\bf b}\times {\bf c}$ is the scalar area of this parallelogram times the component of ${\bf a}$ in the direction of its normal. It follows that ${\bf a}\cdot {\bf b}\times {\bf c}$ is the volume of the parallelepiped defined by vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$ (see Fig. 10). This volume is independent of how the triple product is formed from ${\bf a}$, ${\bf b}$, and ${\bf c}$, except that

$${\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}.$$ (44)

So, the ``volume'' is positive if ${\bf a}$, ${\bf b}$, and ${\bf c}$ form a right-handed set (i.e., if ${\bf a}$ lies above the plane of ${\bf b}$ and ${\bf c}$, in the sense determined from the right-hand grip rule by rotating ${\bf b}$ onto ${\bf c}$) and negative if they form a left-handed set. The triple product is unchanged if the dot and cross product operators are interchanged:

$${\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}.$$ (45)

The triple product is also invariant under any cyclic permutation of ${\bf a}$, ${\bf b}$, and ${\bf c}$,

$${\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b},$$ (46)

but any anti-cyclic permutation causes it to change sign,

$${\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}.$$ (47)

The scalar triple product is zero if any two of ${\bf a}$, ${\bf b}$, and ${\bf c}$ are parallel, or if ${\bf a}$, ${\bf b}$, and ${\bf c}$ are co-planar.

If ${\bf a}$, ${\bf b}$, and ${\bf c}$ are non-coplanar, then any vector ${\bf r}$ can be written in terms of them:

$${\bf r} = \alpha {\bf a} + \beta {\bf b} + \gamma {\bf c}.$$ (48)

Forming the dot product of this equation with ${\bf b}\times {\bf c}$, we then obtain

$${\bf r} \cdot {\bf b} \times{\bf c} = \alpha {\bf a}\cdot{\bf b} \times{\bf c},$$ (49)

so

$$\alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}.$$ (50)

Analogous expressions can be written for $\beta$ and $\gamma$. The parameters $\alpha$, $\beta$, and $\gamma$ are uniquely determined provided ${\bf a}\cdot{\bf b} \times{\bf c} \neq 0$: i.e., provided that the three basis vectors are not co-planar.