1.5 Projection

What is projection

• Much like the name you are projecting a vector onto another vector

• Formula

  • The projection of $\vec{x}$ onto $\vec{v}=/=\vec{0}$

$$pro{j}_{\vec{v}}(\vec{x})=\left(\frac{\vec{x}\ast \vec{v}}{|{\vec{v}}^2|}\right)\vec{v}$$

• the peredicular part of $\vec{x}$ is defiend by

$$per{p}_{\vec{v}}(\vec{x})=\vec{x}-pro{j}_{\vec{v}}(\vec{x})$$

Ex 1

• Find $(1,2,3)={\vec{x}}_1+{\vec{x}}_2,$where ${\vec{x}}_1$ is parallel to $(1,1,1)$ and ${\vec{x}}_2$is perpendicular to $(1,1,1)$

$${\vec{x}}_1=pro{j}_{(1,1,1)}(1,2,3)=\left(\frac{(1,2,3)\ast (1,1,1)}{|(1,1,1{)}^2|}\right)\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]⟹\left[\begin{array}{c}2\\ 2\\ 2\end{array}\right]$$ $${\vec{x}}_2=\vec{x}-{\vec{x}}_1=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]-\left[\begin{array}{c}2\\ 2\\ 2\end{array}\right]=\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$$

Ex2

• what is the projection of (2,3) onto (1,-1)? make a sketch

$$\begin{array}{r}\vec{x}=(2,3),\vec{v}=(1,-1)\\ pro{j}_{\vec{v}}(\vec{x})⟹\left[\begin{array}{c}-1/2\\ 1/2\end{array}\right]\end{array}$$