Projection

Definition of Projection

Take two vectors in $R^n$, $\mathbf{u}$ and $\mathbf{v}$.

The projection of $\mathbf{u}$ onto $\mathbf{v}$ is the vector paralell to $\mathbf{v}$ that's magnitude is component of $\mathbf{u}$ in the direction of $\mathbf{v}$.

One way to visualize this is to shine a flashlight from a direction perpendicular to $\mathbf{v}$, with $\mathbf{u}$ between the flashlight and $\mathbf{v}$ (pretend the vectors are opaque). The shadow the flashlight casts onto $\mathbf{v}$ will be the projection of $\mathbf{u}$ onto $\mathbf{v}$.

The notation and formula for projection of $\mathbf{u}$ onto $\mathbf{v}$ is:

$${\mathrm{proj}}_{\mathbf{v}}\mathbf{u}=\frac{\mathbf{u}\cdot \mathbf{v}}{||\mathbf{v}|{|}^2}\mathbf{v}$$

Component

The length of the projection of $\mathbf{u}$ in the direction of $\mathbf{v}$ is called the component of $\mathbf{u}$ in the direction of $\mathbf{v}$ and is written as:

$${\mathrm{comp}}_{v}u=||{\mathrm{proj}}_{\mathbf{v}}\mathbf{u}||=\frac{\mathbf{u}\cdot \mathbf{v}}{||\mathbf{v}||}$$

Orthogonal

The orthogonal projection of $\mathbf{u}$ onto $\mathbf{v}$ is given as:

$${\mathrm{orth}}_{\mathbf{v}}\mathbf{u}=\mathbf{u}-{\mathrm{proj}}_{\mathbf{v}}\mathbf{u}$$

This is the component of $\mathbf{u}$ that is orthogonal to $\mathbf{v}$.