Properties of the Dot Product of Vectors
What is the Dot Product?
The dot product (also called the scalar product) is an operation that takes two vectors and returns a single number (a scalar). For two vectors $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$:
$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$
Equivalently, using the angle $\theta$ between them:
$$\vec{a} \cdot \vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta$$
The Key Properties
The dot product obeys several important algebraic properties. Understanding these lets you manipulate vector expressions just like you manipulate numbers in algebra.