Properties of Dot Product

Theorem

Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the real Euclidean space $\R^n$.

Let $c$ be a real scalar.

The dot product has the following properties:

Dot Product with Self is Non-Negative

Let $\mathbf u$ be a vector in the real Euclidean space $\R^n$.

Then:

$\mathbf u \cdot \mathbf u \ge 0$

where $\cdot$ denotes the dot product operator.

Dot Product with Self is Zero iff Zero Vector

$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$

Dot Product Operator is Commutative

$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$

Dot Product Operator is Bilinear

$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$

That last result is often broken down into two less powerful ones:

Dot Product Distributes over Addition

$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$

Dot Product Associates with Scalar Multiplication

$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$