Projection

Let $X$ be a set. The projection of $\vec{v}$ onto $X$, $\text{proj}_x\vec{v}$, is the closest point in $X$ to $\vec{v}$

Find $\text{proj}_{\vec{u}_t}\vec{v}$ by minimizing $||\vec{u}_t-\vec{v}||$

For lines and planes: Vector from the projection to the original point is a normal vector.

• If $X$ is a line or plane, $\vec{v}-\text{proj}_X\vec{v}$ is a normal vector for $X$ (Orthogonal to $\vec{d}$)

Undefined Cases

• If two points in $X$ are the same distance to $\vec{v}$, $\text{proj}_X\vec{v}$ is undefined
• If $X$ is an open set (E.g. $(0,1]$), $\text{proj}_X\vec{v}$ might be undefined

Vector Components

Let $\vec{u}$ and $\vec{v}≠\vec{0}$, the vector component of $\vec{u}$ in the $\vec{v}$ direction is $\text{vcomp}_{\vec{v}}\vec{u}$

• $\vec{u}-\text{vcomp}_{\vec{v}}\vec{u}$ is orthogonal to $\vec{v}$
• $\text{vcomp}_{\vec{b}}\vec{a}=(\frac{\vec{b}\cdot\vec{a}}{\vec{b}\cdot\vec{b}})\vec{b}$
• $(\vec{u}-\text{vcomp}_{\vec{v}}\vec{u})\cdot\vec{u} =0$
• $\text{vcomp}_{\vec{v}}\vec{u}=k\vec{v}$ for some $k\inℝ$
• $\text{proj}{\text{span}\{\vec{v}\}}\vec{u}=\text{vcomp}{\vec{v}}\vec{u}$