Projections

The projection of one vector onto another, its length, the component of a vector perpendicular to another, resolving a vector into perpendicular parts, and the distance from a point to a line.

The projection

The projection of $\underset{\sim}{u}$ onto $\underset{\sim}{v}$ is the "shadow" of $\underset{\sim}{u}$ on the line of $\underset{\sim}{v}$, when light shines perpendicular to that line.

Projection of u onto v

$$\operatorname{proj}_{\underset{\sim}{v}}\underset{\sim}{u} = \frac{\underset{\sim}{u} \cdot \underset{\sim}{v}}{\underset{\sim}{v} \cdot \underset{\sim}{v}}\,\underset{\sim}{v} \qquad \text{length} = \frac{|\underset{\sim}{u} \cdot \underset{\sim}{v}|}{|\underset{\sim}{v}|}$$

The perpendicular component

Every vector splits into a part along $\underset{\sim}{v}$ and a part perpendicular to it: $$\underset{\sim}{u} = \operatorname{proj}_{\underset{\sim}{v}}\underset{\sim}{u} + \big(\underset{\sim}{u} - \operatorname{proj}_{\underset{\sim}{v}}\underset{\sim}{u}\big).$$

Example 1●●

$\underset{\sim}{u} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\underset{\sim}{v} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$. Resolve $\underset{\sim}{u}$ into parts parallel and perpendicular to $\underset{\sim}{v}$.

$\underset{\sim}{u} \cdot \underset{\sim}{v} = 10$ and $\underset{\sim}{v} \cdot \underset{\sim}{v} = 10$, so the projection is $1 \times \underset{\sim}{v} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

The perpendicular part is $\begin{pmatrix} 2 \\ 4 \end{pmatrix} - \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$.

Check: $\begin{pmatrix} -1 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 1 \end{pmatrix} = 0$.

Example 2: distance to a line●●●

Find the distance from $P(2, 4)$ to the line through the origin in the direction $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

The distance is the length of the perpendicular part of $\overrightarrow{OP}$, which Example 1 found to be $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$.

Distance $= \sqrt{1 + 9} = \sqrt{10}$.

Flashcards

$operatorname{proj}_{underset{sim}{v}}underset{sim}{u}$

$\dfrac{\underset{\sim}{u} \cdot \underset{\sim}{v}}{\underset{\sim}{v} \cdot \underset{\sim}{v}}\,\underset{\sim}{v}$

Length of the projection

$\dfrac{|\underset{\sim}{u} \cdot \underset{\sim}{v}|}{|\underset{\sim}{v}|}$

Perpendicular component

$\underset{\sim}{u} - \operatorname{proj}_{\underset{\sim}{v}}\underset{\sim}{u}$

Practice

Find the projection of $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$. Enter it as $x, y$.
Find the component of $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ perpendicular to $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$. Enter it as $x, y$.
Find the distance from $P(2, 4)$ to the line through the origin in the direction $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.
Find the length of the projection of $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$ onto $\underset{\sim}{i}$.
Find the length of the projection of $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ onto $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$.
Find the length of the projection of $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ onto $\begin{pmatrix} -1 \\ 2 \end{pmatrix}$.
The projection of $\underset{\sim}{u}$ onto $\underset{\sim}{v}$ is $k\underset{\sim}{v}$, where $\underset{\sim}{u} \cdot \underset{\sim}{v} = 18$ and $|\underset{\sim}{v}| = 3$. Find $k$.
The projection of $\underset{\sim}{u}$ onto $\underset{\sim}{v}$ is:
If $\underset{\sim}{u} \perp \underset{\sim}{v}$, the projection of $\underset{\sim}{u}$ onto $\underset{\sim}{v}$ is: