Scalar Triple Product

The scalar triple product is

$\displaystyle \boxed{[\mathbf{u},\mathbf{v},\mathbf{w}] =(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}.} \tag{1}$

Its absolute value equals the volume of the parallelepiped generated by the three vectors:

$\displaystyle V=\vert[\mathbf{u},\mathbf{v},\mathbf{w}]\vert.$

Its sign records orientation relative to a right-handed basis.

Figure 18, modernized: scalar triple product as oriented parallelepiped volume.

The scalar triple product is invariant under cyclic permutation:

$\displaystyle (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w} =(\mathbf{v}\times\mathbf{w})\cdot\mathbf{u} =(\mathbf{w}\times\mathbf{u})\cdot\mathbf{v},$

and changes sign under exchange of any two vectors.

In Cartesian components,

$\displaystyle \boxed{ (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}= \begin{vmatrix} u_x&u_y&u_z\ v_x&v_y&v_z\ w_x&w_y&w_z \end{vmatrix}.} \tag{2}$

Therefore nonzero vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ are coplanar exactly when their scalar triple product vanishes.

Example 1: volume

For

$\displaystyle A=(-3,1,2),\ B=(-1,0,-2),\ C=(2,1,4),\ D=(2,-3,1),$

Brand obtains

$\displaystyle (\overrightarrow{AB}\times\overrightarrow{AC})\cdot \overrightarrow{AD}=81.$

Thus the parallelepiped volume is $81$

, and the tetrahedron volume is $81/6=13.5$

Example 2: line-plane intersection

If a line is

$\displaystyle \mathbf r=\mathbf{r}_A+\lambda\mathbf d$

and a plane through $C,D,E$

has normal

$\displaystyle \mathbf n=(\mathbf r_D-\mathbf r_C)\times (\mathbf r_E-\mathbf r_C),$

then the intersection parameter is

$\displaystyle \boxed{\lambda= \frac{(\mathbf r_C-\mathbf{r}_A)\cdot\mathbf n} {\mathbf d\cdot\mathbf n}.}$

For Brand's data $A=(1,2,1)$

, this gives $\lambda=2$ and

$\displaystyle P=(3,0,3).$

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

J. R. Taylor, Classical Mechanics, University Science Books, 2005.
D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

This article is a modernized restatement of the corresponding Public Domain article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.

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