The dot product or scalar product is defined as

Geometric interpretation

Using a geometric interpretation allows us to find the angle between two vectors because

It is also useful to note that if the two vectors are perpindicular, their dot product is zero since

The dot product is positive, zero, or negative according as the angle is acute, right, or obtuse. For nonzero vectors,

Also

The dot product is commutative:

For a unit vector ê,

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