The dot product is an algebraic operation that takes 2 equally sized vectors and returns a single scalar number. The dot product is usually denoted with a symbol.

The dot product can be defined as follows:

Where is the size of the vectors.

The dot product can also be defined geometrically

where is the angle between the two vectors

Properties

Two vectors are orthogonal iff their dot product is 0. This is evident from the term in the 2d geometric interpretation.

There are also more general properties:

• Commutativity:
• Distributivity with Addition:
• Distributivity with Scalar Multiplication:
• Non-Associativity: is undefined

Self Dot Product

Given a vector it's dot product with itself is just the square of it's magnitude:

This is because the angle as it's aligned with itself so it just becomes .