# Point A is at (-1 ,4 ) and point B is at (-3 ,7 ). 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?

Mar 14, 2016

Distance has increased by $4.456$

#### Explanation:

A point $\left(x , y\right)$ rotated clockwise $3 \frac{\pi}{2}$ means rotated anticlockwise $\frac{\pi}{2}$ and hence new coordinates are $\left(- y , x\right)$.

Hence the coordinates of point $A$ $\left(- 1 , 4\right)$ will become $\left(- 4 , - 1\right)$.

The distance between point $A$ $\left(- 1 , 4\right)$ and point $B$ $\left(- 3 , 7\right)$ is

$\sqrt{{\left(\left(- 3\right) - \left(- 1\right)\right)}^{2} + {\left(7 - 4\right)}^{2}} = \sqrt{{\left(- 2\right)}^{2} + {3}^{2}}$

= $\sqrt{4 + 9} = \sqrt{13} = 3.606$

The distance between rotated point $A$ $\left(- 4 , - 1\right)$ and point $B$ $\left(- 3 , 7\right)$ is

$\sqrt{{\left(\left(- 3\right) - \left(- 4\right)\right)}^{2} + {\left(7 - \left(- 1\right)\right)}^{2}} = \sqrt{{1}^{2} + {8}^{2}}$

= $\sqrt{1 + 64} = \sqrt{65} = 8.062$

Hence, distance has increased by $8.062 - 3.606 = 4.456$