Definition:Dot Product/Real Euclidean Space

Definition
Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.

The dot product of $\mathbf a$ and $\mathbf b$ is defined as:
 * $\mathbf a \cdot \mathbf b = \left\Vert{\mathbf a}\right\Vert \left\Vert{\mathbf b}\right\Vert \cos \angle \mathbf a, \mathbf b$

where:
 * $\left\Vert{\mathbf a}\right\Vert$ denotes the length of $\mathbf a$
 * $\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.

Also see

 * Equivalence of Definitions of Dot Product