
# Angles between vectors¶

Consider two vectors $$\vec{w}$$ and $$\vec{v}$$.

We already know from Vector projection that the vector projection of $$\vec{w}$$ onto $$\vec{v}$$ is $$c \vec{v}$$ where:

$c = \frac{\vec{w} \cdot \vec{v}}{\VL{v}^2}$

We know from the definition of projection that $$c \vec{v}$$ and $$\vec{w} - c \vec{v}$$ form a right angle.

If the angle between $$\vec{v}$$ and $$\vec{w}$$ is $$\alpha$$, then:

$\VL{w} cos(\alpha) = \L{ c \vec{v} } = c \VL{v} = \frac{\vec{w} \cdot \vec{v}}{\VL{v}}$

and:

$\VL{v} \VL{w} cos(\alpha) = \vec{w} \cdot \vec{v}$