Projections

Topics:

• Be able to decompose vectors into orthogonal components.
  • Know how to compute the orthogonal projection of one vector onto another.

Decomposing Vectors into Orthogonal Components

or Finding Projections

study this

consider two vectors $a=\overrightarrow{PQ}$ and $b=\overrightarrow{PR}$

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Draw a perpendicular line from $R$ to $a$ or its line of direction to find a point $S$. The vector $\overrightarrow{PS}$ is called the orthogonal projection of $b$ onto $a$ or the vector projection of $b$ onto $a$ and is denoted \text{proj}_\vec{a}\vec{a} and the length of the vector projection is called the scalar projection and is denoted ${\text{comp}}_{\text{a}}\text{b}$.

To find ${\text{proj}}_{\vec{a}}\vec{b}{\text{ first find comp}}_{\vec{a}}\vec{b}$

\text{comp}_\vec{a}\vec{b} = \|\vec{b}\| \cos \theta = \frac{\|\vec{a}\| \|\vec{b}\| \cos \theta}{\|\vec{a}\|} = \underbrace{{\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|}}}_{{\text{no }\theta\text{!}}}

$\text{Now we find a vector in the direction of }\vec{a}\text{ with that length:}$

$${\text{proj}}_{\vec{a}}\vec{b}=\left(\frac{\vec{a}\cdot \vec{b}}{\Vert \vec{a}\Vert }\right)\cdot \frac{\vec{a}}{\underset{{}_{\text{unit!}}}{\underbrace{\Vert \vec{a}\Vert }}}=\left(\frac{\vec{a}\cdot \vec{b}}{\vec{a}\cdot \vec{a}}\right)\vec{a}$$

$\text{It is common to complete the decomposition of }\vec{b}=\overrightarrow{PR}\text{ into two vectors:}$ $\text{One parallel to }\vec{a}{\text{ [that is, proj}}_{\vec{a}}\vec{b}\text{] and the other orthogonal to }\vec{a}\text{:}$

$${\text{proj}}_{\vec{a}}\vec{b}+\vec{w}=\vec{b}$$ $$\vec{w}=\vec{b}-{\text{proj}}_{\vec{a}}\vec{b}$$

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