# Point A is at (6 ,2 ) and point B is at (3 ,-8 ). Point A is rotated pi/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?

Feb 21, 2018

The new point $A '$ is $\left(2 , - 6\right)$, and the distance changed by 8.204 units.

#### Explanation:

There is a formal method of doing this, and there is an easier way for simpler problems. I present the formal method first.

Given a point $\left(x , y\right)$, rotating it around the origin by angle $\theta$ results in the coordinate $\left(x ' , y '\right)$:
$x ' = x \cos \theta - y \sin \theta$
$y ' = y \cos \theta + x \sin \theta$

Now, imagining you haven't taken trig yet (this is overkill for a 90 degree turn anyways), here's a (perhaps) more intuitive method.

Imagine taking the entire coordinate axis and rotating it 90 degrees clockwise about the origin in your head. The positive x-axis is now where the negative y-axis used to be. The positive y-axis is now where the positive x-axis used to be, and so on.

The point used to be at $\left(6 , 2\right)$. It used to be 6 units to the right and 2 units up. However, since you turned it, it's now 6 units down and 2 units right. Make sure you can visualize this in your head, it's a useful skill. It might help to physically draw the coordinate axis, draw the point, and rotate the paper.

Down means negative y-axis, right means positive x-axis. The new coordinates for point $A '$ is $\left(2 , - 6\right)$.

To determine how much the distance changed, we simply take the distances between $A$ and $B$, and $A '$ and $B$.

$d \left(A , B\right) = \sqrt{{\left(6 - 3\right)}^{2} + {\left(2 - \left(- 8\right)\right)}^{2}} = \sqrt{109} \approx 10.440$
$d \left(A ' , B\right) = \sqrt{{\left(2 - 3\right)}^{2} + {\left(- 6 - \left(- 8\right)\right)}^{2}} = \sqrt{5} \approx 2.236$
$| d \left(A , B\right) - d \left(A ' , B\right) | = 8.204$

$\therefore$ the new coordinates for point $A '$ is $\left(2 , - 6\right)$ and the distance between points $A$ and $B$ changed by $8.204$ units.

$\square$