vector_projection.md

Vector projection

This here page follows the discussion in this Khan academy video on projection. Please watch that video for a nice presentation of the mathematics on this page.

For the video and this page, you will need the definitions and mathematics from on vectors.

Start

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

We can scale $\vec{v}$ with a scalar $c$. By choosing the correct $c$ we can create any \vector on the infinite length dotted line in the diagram. $c \vec{v}$ defines this infinite line.

We're going to find the projection of $\vec{w}$ onto $\vec{v}$, written as:

$$ \mathrm{proj}_\vec{v}\vec{w} $$

The projection of $\vec{w}$ onto $\vec{v}$ is a vector on the line $c \vec{v}$. Specifically it is $c \vec{v}$ such that the line joining $\vec{w}$ and $c \vec{v}$ is perpendicular to $\vec{v}$.

Why is it called projection?

Imagine a light source, parallel to $\vec{v}$, above $\vec{w}$. The light would cast rays perpendicular to $\vec{v}$.

$\mathrm{proj}_\vec{v}\vec{w}$ is the shadow cast by $\vec{w}$ on the line defined by $\vec{v}$.

Calculating the projection

The \vector connecting $\vec{w}$ and $c \vec{v}$ is $\vec{w} - c \vec{v}$.

We want to find $c$ such that $\vec{w} - c \vec{v}$ is perpendicular to $\vec{v}$.

Two perpendicular vectors have a vector dot product of zero (see on vectors), and so:

$$(\vec{w} - c \vec{v}) \cdot \vec{v} = 0$$

By distribution over addition of dot products:

$$\begin{aligned} (\vec{w} - c \vec{v}) \cdot \vec{v} = 0 \implies \\ \vec{w} \cdot \vec{v} - c \vec{v} \cdot \vec{v} = 0 \implies \\ \frac{\vec{w} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} = c \end{aligned}$$

Because $| \vec{v} | = \sqrt(\vec{v} \cdot \vec{v})$:

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

So:

$$\mathrm{proj}_\vec{v}\vec{w} = \frac{\vec{w} \cdot \vec{v}}{\| \vec{v} |^2} \vec{v}$$

We can also write the projection in terms of the unit \vector defined by $\vec{v}$:

$$\begin{aligned} \hat{u} \triangleq \frac{\vec{v}}{\| \vec{v} |} \implies \\ \mathrm{proj}_\vec{v}\vec{w} = \frac{\vec{w} \cdot \vec{v}}{\| \vec{v} |} \vec{u} \end{aligned}$$

$\frac{\vec{w} \cdot \vec{v}}{| \vec{v} |}$ is called the scalar projection of $\vec{w}$ onto $\vec{v}$.

Also see

• wikipedia on \vector projection
• On vectors