Question

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?

Answer 1

The new point `A'` is `(2,-6)`, 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 `(x,y)`, rotating it around the origin by angle `theta` results in the coordinate `(x',y')`:
`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 `(6,2)`. 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 `(2, -6)`.

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

`d(A,B) = sqrt((6-3)^2+(2-(-8))^2) = sqrt(109) approx 10.440`
`d(A',B) = sqrt((2-3)^2+(-6-(-8))^2) = sqrt(5) approx 2.236`
`|d(A,B) - d(A',B)| = 8.204`

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

`square`