# Dot product

THEOREM

The dot product is commutative in $$\mathbb{R}^2$$.

$$\forall \mathbf{u},\mathbf{v} \in \mathbb{R}^2 : \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$$

Proof available

THEOREM

The dot product is commutative.

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

Proof available

THEOREM

The dot product is left-distributive over addition.

$$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{uv} + \mathbf{uw}$$

Proof available

THEOREM

The dot product is right-distributive over addition.

$$(\mathbf{v} + \mathbf{w}) \cdot \mathbf{u} = \mathbf{vu} + \mathbf{wu}$$

Proof available

THEOREM    $$\mathbf{u} \cdot \mathbf{u} \ge 0$$

Proof available

THEOREM    $$\mathbf{u} \cdot \mathbf{u} = 0 \Leftrightarrow \mathbf{u} = \mathbf{0}$$

Proof available

THEOREM    $$(c\mathbf{u} + \mathbf{v}) \cdot w = c(\mathbf{u} \cdot \mathbf{w}) + (\mathbf{v} \cdot \mathbf{w})$$

Proof available

THEOREM    $$(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})$$

Proof available