Products of Two Vectors

Two distinct vector products are fundamental in three-dimensional mechanics. The dot product

returns a scalar and measures directional alignment. The cross product

returns a vector perpendicular to both inputs and encodes oriented area.

These products appear throughout mechanics:

• work: W = ∫ F ⋅ dr,
• power: P = F ⋅ v,
• torque: τ = r × F,
• angular momentum: L = r × p.

Vector division by a vector is not part of ordinary Euclidean vector algebra. Division by a nonzero scalar remains ordinary scalar multiplication by its reciprocal.

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 article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.