Angles Between Vectors

14. Angles Between Vectors

The scalar $‖ \vec{𝑢} ‖$ is the length of the vector $\vec{𝑢}$

Say $\vec{𝑢}, \vec{𝑣} \in ℝ^{𝑛}$

Law of Cosines

𝑐^{2} = 𝑎^{2} + 𝑏^{2} - 2 𝑎 𝑏 \cdot cos (𝐶)

Where:

\begin{aligned} ‖ \vec{𝑢} - \vec{𝑣} ‖^{2} & = ‖ \vec{𝑢} ‖^{2} + ‖ \vec{𝑣} ‖^{2} - 2 ‖ \vec{𝑢} ‖ ‖ \vec{𝑣} ‖ \cdot cos (Θ) \\ (\vec{𝑢} - \vec{𝑣}) \cdot (\vec{𝑢} - \vec{𝑣}) & = \\ \vec{𝑢} \cdot \vec{𝑢} - \vec{𝑢} \cdot \vec{𝑣} - \vec{𝑣} \cdot \vec{𝑢} + \vec{𝑣} \cdot \vec{𝑣} & = \\ ‖ \vec{𝑢} ‖^{2} - 2 (\vec{𝑢} \cdot \vec{𝑣}) + ‖ \vec{𝑣} ‖ & = ‖ \vec{𝑢} ‖^{2} + ‖ \vec{𝑣} ‖^{2} - 2 ‖ \vec{𝑢} ‖ ‖ \vec{𝑣} ‖ \cdot cos (Θ) \end{aligned}

\vec{𝑢} \cdot \vec{𝑣} = ‖ \vec{𝑢} ‖ ‖ \vec{𝑣} ‖ \cdot cos (Θ)

\frac{\vec{𝑢} \cdot \vec{𝑣}}{‖ \vec{𝑢} ‖ ‖ \vec{𝑣} ‖} = cos (Θ)

Θ = arccos (\frac{\vec{𝑎} \cdot \vec{𝑏}}{‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖})

So, if $\vec{𝑢}$ is a scalar multiple of $\vec{𝑣}$ ( $\vec{𝑢} = 𝑐 \vec{𝑣}$ ) where $𝑐 > 0$ , then $Θ = 0 °$

And, if $\vec{𝑢}$ is a scalar multiple of $\vec{𝑣}$ ( $\vec{𝑢} = 𝑐 \vec{𝑣}$ ) where $𝑐 < 0$ , then $Θ = 180 °$

$\vec{𝑢}$ and $\vec{𝑣}$ are perpendicular if the angle $Θ$ between them is $90 °$

\begin{aligned} \vec{𝑢} \cdot \vec{𝑣} & = ‖ \vec{𝑢} ‖ ‖ \vec{𝑣} ‖ \cdot cos (90) \\ \vec{𝑢} \cdot \vec{𝑣} & = 0 \end{aligned}

If $\vec{𝑢}$ and $\vec{𝑣}$ are perpendicular, then $\vec{𝑢} \cdot \vec{𝑣} = 0$

If $\vec{𝑢}$ and $\vec{𝑣}$ are non-zer0 and $\vec{𝑢} \cdot \vec{𝑣} = 0$ , then they are perpendicular

If $\vec{𝑢} \cdot \vec{𝑣} = 0$ then $\vec{𝑢}$ and $\vec{𝑣}$ are orthogonal.