Dot Products and Length

Dot products are done like so,

$$\vec{u}\cdot \vec{v}={\vec{u}}^T\vec{v}=\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_n\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]=u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n$$

Properties

$(\vec{v}+\vec{w})\cdot \vec{u}=\vec{v}\cdot \vec{u}+\vec{w}\cdot \vec{u}$ $(c\vec{v})\cdot \vec{u}=c(\vec{v}\cdot \vec{u})$ $\vec{v}\cdot \vec{u}=\vec{u}\cdot \vec{v}$ $\vec{u}\cdot \vec{u}\ge 0$ $\vec{u}\cdot \vec{u}=0⟺\vec{u}=0$

Length

$||\vec{u}||=\sqrt{\vec{u}\cdot \vec{u}}$

Properties

$||c\vec{v}\Vert |=|c|\cdot ||\vec{v}||$

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