Question

Point A is at `(4 ,-2 )` and point B is at `(2 ,-3 )`. 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

New coordinate: `(-2, -4)`
Difference in distance: `sqrt(5)-1`

Explanation:

Finding the new coordinate
With the `x` -coordinate positive and `y` -coordinate negative the point `A` used to lie in the fourth quadrant of the Cartesian plane (the bottom-right corner); rotating it clockwise by `pi/2` (or equivalently by `90^"o"` ) would move it to the third quadrant (the bottom-left corner) such that both its `x` and `y` -coordinates are now positive.

Rotating a point with respect to the origin by an odd integer multiple of `pi/2` or `90^"o"` would swap its `x` and `y` -coordinates. So now point `A` would have coordinates `(-2, -4)`.

Finding and comparing the distance between `A` and `B`
Now that we have got the coordinates of `A` before and after the rotation, so we can apply the Pythagorean theorem in the Cartesian to find the distance between point `A` and `B`:

Before the rotation:
`A_i B = sqrt((x_(A_i)- x_B)^2+(y_(A_i)- y_B)^2)=sqrt((4-2)^2+(-2-(-3))^2)=sqrt(5)`

After the rotation:
`A_fB = sqrt((x_(A_f)- x_B)^2+(y_(A_f)- y_B)^2)=sqrt((-2-2)^2+(-4-(-3))^2)=1`

Therefore the difference would be
`abs(A_i B-A_f B)=sqrt(5)-1`

References/ See also:
"Distance Between Two Points", Massey University,
http://mathsfirst.massey.ac.nz/Algebra/PythagorasTheorem/pythapp.htm
Wikipedia contributors. "Quadrant (plane geometry)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 22 Feb. 2018. https://en.wikipedia.org/wiki/Quadrant_(plane_geometry) 14 Mar. 2018.