Scalar Triple Product

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 multiplied by 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 Figure A.12. This volume is independent of how the triple product is formed from ${\bf a}$, ${\bf b}$, and ${\bf c}$, except that

$${\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}.$$ (1.53)

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 a right-hand circulation 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,

$${\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}.$$ (1.54)

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

$${\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b},$$ (1.55)

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

$${\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}.$$ (1.56)

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 coplanar.

Figure A.12: A vector parallelepiped.

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

$${\bf r} = \alpha \,{\bf a} + \beta\,{\bf b} + \gamma\, {\bf c}.$$ (1.57)

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

$${\bf r} \cdot {\bf b} \times{\bf c} = \alpha\, {\bf a}\cdot{\bf b} \times{\bf c},$$ (1.58)

so

$$\alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}.$$ (1.59)

Analogous expressions can be written for $\beta$ and $\gamma$. The parameters $\alpha$, $\beta$, and $\gamma$ are uniquely determined provided ${\bf a}\cdot{\bf b} \times{\bf c} \neq 0$; that is, provided that the three vectors are non-coplanar.