Finding the angle of rotation given the slope of the line connecting two rotated points.

I am trying to apply certain special affine transformations to sets in 2-space. One of the simplest problems I am working with: I have original points (3,8) and (6,1) that lie on a line with slope m = -7/3 . I would like to rotate these two points (counterclockwise through the origin) so that the rotated points lie on a new line with slope m = -1 .

This is easy if one can find the angle of rotation but I am having trouble even with mupad. I tried rotating the original two points with the 2x2 matrix

cos(w) -sin(w)

sin(w) cos(w)

and then solving for w in mupad but the output (right below) appears to show that the set of rotation angles are all complex valued which seems wrong.

(PI*k - (ln(21/29 + (20*I)/29)*I)/2, k, Z_)

If I can find a real value for w in either radians or degrees I think I can proceed. Am I right that this problem has an easy solution I am simply missing?

Answer 1

[Matt J]

The rotation angle is easy to calculate,

   w = atand(-1) - atand(-7/3)  %clockwise rotation angle in degrees

It depends only on the slopes, not on where the center of rotation is.