The cross product (or vector product) of A = (A_x,A_y,A_z) and B = (B_x,B_y,B_z) is

Geometrically, its magnitude is

where 𝜃 is the smaller angle between A and B. Its direction is perpendicular to the plane determined by the two vectors and is selected by the right-hand rule:

It is often convenient to remember the cross product through the determinant mnemonic

Vector Product (Cross Product)

For vectors A,B ∈ ℝ³ separated by the angle 𝜃, one may summarize the geometric definition as

(1)

Its magnitude equals the area of the parallelogram spanned by A and B:

For nonzero vectors,

The cross product is anti-commutative:

It is a specifically three-dimensional Euclidean operation in this form.

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.