scalar triple product

Scalar Triple Product

The scalar triple product is

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

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

The scalar triple product is invariant under cyclic permutation:

and changes sign under exchange of any two vectors.

In Cartesian components,

Therefore nonzero vectors u,v,w are coplanar exactly when their scalar triple product vanishes.

Example 1: volume

For

Brand obtains

Thus the parallelepiped volume is 81, and the tetrahedron volume is 81∕6 = 13.5.

Example 2: line-plane intersection

If a line is

and a plane through C,D,E has Normal

then the intersection parameter is

For Brand’s data A = (1, 2, 1), B = (2, 1, 2), C = (0,−4, 4), D = (2,−2, 2), E = (4, 1, 2), this gives λ = 2 and

Modern notation references

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

1. J. R. Taylor, Classical Mechanics, University Science Books, 2005.
2. D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
3. 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.