The cross product or vector product is defined by

Like the dot product, it is useful to look at its geometric definition and properties. Instead of the cosine of the angle between the two vectors the cross product is defined geometrically as

It is important to see that the unit vector n is normal to the plane defined by the two vectors with the direction determined by the right hand rule.

It can be easier to remember the definition of the cross product with the determinant formulation

Vector Product (Cross Product)

For vectors A,B ∈ ℝ³ separated by the smaller angle 𝜃, the cross product is defined by

where n is the unit normal selected by the right-hand rule.

Its magnitude equals the area of the parallelogram generated by the vectors:

For nonzero vectors,

The cross product is anticommutative:

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.