# A line segment has endpoints at (5 ,9 ) and (8 ,7 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -8 , and reflected about the y-axis, what will the line segment's new endpoints be?

Mar 31, 2016

$A ' ' ' = \left[\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right] \left[\begin{matrix}- 9 \\ - 3\end{matrix}\right] = \left[\begin{matrix}9 \\ - 3\end{matrix}\right]$

$B ' ' ' = \left[\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right] \left[\begin{matrix}- 7 \\ 0\end{matrix}\right] = \left[\begin{matrix}7 \\ 0\end{matrix}\right]$

#### Explanation:

Given : Two points $A \left(5 , 9\right)$ and $B \left(8 , 7\right)$. Rotate by $\frac{\pi}{2}$, translate vertically -8 and reflected about y-axis
Required : New end-point of the segment $A '$ and $B '$
Solution Strategy : a) Rotate b) Translated Vertically c) Reflect

a) Rotate using Rotation Matrix :
$R \left(\frac{\pi}{2}\right) = \left[\begin{matrix}\cos \left(\frac{\pi}{2}\right) & - \sin \left(\frac{\pi}{2}\right) \\ \sin \left(\frac{\pi}{2}\right) & \cos \left(\frac{\pi}{2}\right)\end{matrix}\right] = \left[\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right]$
$A ' = \left[\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right] \left[\begin{matrix}5 \\ 9\end{matrix}\right] = \left[\begin{matrix}- 9 \\ 5\end{matrix}\right]$

$B ' = \left[\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right] \left[\begin{matrix}8 \\ 7\end{matrix}\right] = \left[\begin{matrix}- 7 \\ 8\end{matrix}\right]$

b) Translate Vertically :
Translation Operation of vector, $P$ by ${\delta}_{x , y}$ is given by:
$\vec{P '} = \vec{P} + {\vec{\delta}}_{x , y}$ where
${\delta}_{x , y} = \left[\begin{matrix}{\delta}_{x} \\ {\delta}_{y}\end{matrix}\right]$ thusd
$A ' ' = \left[\begin{matrix}- 9 \\ 5\end{matrix}\right] + \left[\begin{matrix}0 \\ - 8\end{matrix}\right] = \left[\begin{matrix}- 9 \\ - 3\end{matrix}\right]$

$B ' ' = \left[\begin{matrix}- 7 \\ 8\end{matrix}\right] + \left[\begin{matrix}0 \\ - 8\end{matrix}\right] = \left[\begin{matrix}- 7 \\ 0\end{matrix}\right] =$

c) Reflection Vertically :
Reflection Matrix about y, is $R {f}_{y} = \left[\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right]$
$A ' ' ' = \left[\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right] \left[\begin{matrix}- 9 \\ - 3\end{matrix}\right] = \left[\begin{matrix}9 \\ - 3\end{matrix}\right]$

$B ' ' ' = \left[\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right] \left[\begin{matrix}- 7 \\ 0\end{matrix}\right] = \left[\begin{matrix}7 \\ 0\end{matrix}\right]$