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- MAT223 Readings 06

Vector Projection

Definition

$\vec{w}$ is a vector. For any $\vec{v}$ , you can write $\vec{v} = {\vec{v}}_{1} + {\vec{v}}_{2}$ where ${\vec{v}}_{1} ∥ \vec{w}$ and ${\vec{v}}_{2} ⊥ \vec{w}$
${\vec{v}}_{1}$ is the projection of $\vec{v}$ onto $\vec{w}$
${\vec{v}}_{1} = {proj}_{\vec{w}} \vec{v}$

Theorem

For any two vectors $\vec{v}$ and $\vec{w}$ , we have ${proj}_{\vec{w}} \vec{v} = \frac{\vec{v} \cdot \vec{w}}{{| \vec{w} |}^{2}} \vec{w}$
${\vec{v}}_{2} = \vec{v} - {proj}_{\vec{w}} \vec{v}$

Example:
- Find ${proj}_{\vec{v}} \vec{w}$
- $\vec{w} = {\vec{w}}_{1} + {\vec{w}}_{2}$
- ${\vec{w}}_{1} ∥ \vec{v}$
- ${\vec{w}}_{2} ⊥ \vec{v}$
- $\vec{v} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$
- $\vec{w} = [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}]$
- ${proj}_{\vec{v}} \vec{w} = \frac{\vec{v} \cdot \vec{w}}{{| \vec{v} |}^{2}} \vec{v}$
- ${proj}_{\vec{v}} \vec{w} = \frac{(1) (2) + (1) (1) + (1) (2)}{1^{2} + 1^{2} + 1^{2}} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$
- ${proj}_{\vec{v}} \vec{w} = \frac{5}{3} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$
- ${proj}_{\vec{v}} \vec{w} = [\begin{matrix} \frac{5}{3} \\ \frac{5}{3} \\ \frac{5}{3} \end{matrix}]$
- ${\vec{w}}_{1} = [\begin{matrix} \frac{5}{3} \\ \frac{5}{3} \\ \frac{5}{3} \end{matrix}]$
- ${\vec{w}}_{2} = \vec{w} - {\vec{w}}_{1}$
  - $= [\begin{matrix} 2 \\ 1 \\ 2 \end{matrix}] - [\begin{matrix} \frac{5}{3} \\ \frac{5}{3} \\ \frac{5}{3} \end{matrix}]$
    - $2 - \frac{5}{3} = \frac{1}{3}$
    - $1 - \frac{5}{3} = - \frac{2}{3}$
  - $= [\begin{matrix} \frac{1}{3} \\ - \frac{2}{3} \\ \frac{1}{3} \end{matrix}]$