# Point A is at (-6 ,1 ) and point B is at (3 ,8 ). Point A is rotated (3pi)/2  clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Jun 24, 2017

The new coordinates are $= \left(- 1 , - 6\right)$ and the distance has changed by $= 3.2$

#### Explanation:

The matrix of a rotation clockwise by $\frac{3}{2} \pi$ about the origin is

$\left(\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right)$

Therefore, the transformation of point $A$ is

$A ' = \left(\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right) \left(\begin{matrix}- 6 \\ 1\end{matrix}\right) = \left(\begin{matrix}- 1 \\ - 6\end{matrix}\right)$

The distance $A B$ is

$= \sqrt{{\left(3 - \left(- 6\right)\right)}^{2} + {\left(8 - 1\right)}^{2}}$

$= \sqrt{81 + 49}$

$= \sqrt{130}$

The distance $A ' B$ is

$= \sqrt{{\left(3 - \left(- 1\right)\right)}^{2} + {\left(8 - \left(- 6\right)\right)}^{2}}$

$= \sqrt{16 + 196}$

$= \sqrt{212}$

The distance has changed by

$= \sqrt{212} - \sqrt{130}$

$= 3.2$