Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

2.1.6 Triple products

Two combinations of three vectors appear repeatedly. The scalar triple product $A \cdot (B \times C)$ equals the determinant whose rows are the components of $A$ , $B$ , $C$ :

A \cdot (B \times C) = A_{x} B_{x} C_{x} A_{y} B_{y} C_{y} A_{z} B_{z} C_{z} . (2.1.12)

(Take the dot product of $A$ with the expanded form of (2.1.8) written for $B \times C$ : the unit vectors of the first row are simply replaced by $A_{x}, A_{y}, A_{z}$ .) Its absolute value is the volume of the parallelepiped with edges $A$ , $B$ , $C$ (Figure 2.1.7): $∣ B \times C ∣$ is the area of the base, and $A \cdot a_{n}$ , the projection of $A$ on the normal of the base, is the height. Since the same volume is obtained with any face as base, the product is unchanged under a cyclic permutation, $A \cdot (B \times C) = B \cdot (C \times A) = C \cdot (A \times B)$ . The vector triple product obeys the "back-cab" rule

A \times (B \times C) = B (A \cdot C) - C (A \cdot B), (2.1.13)

which can be checked by writing out both sides in components.

The scalar triple product as a volume. The base parallelogram spanned by \vec B and \vec C has area |\vec B\times\vec C| and normal \vec a_n along \vec B\times\vec C, and the height is the projection of \vec A on that normal. Volume = base × height =|\vec A\cdot(\vec B\times\vec C)|. — **Figure 2.1.7.** The scalar triple product as a volume. The base parallelogram spanned by $B$ and $C$ has area $∣ B \times C ∣$ and normal $a_{n}$ along $B \times C$ , and the height is the projection of $A$ on that normal. Volume = base × height $= ∣ A \cdot (B \times C) ∣$ .