Rotating a Point

Date: 04/08/2003 at 09:19:04
From: Reggie
Subject: Coordinates on a grid

What is the formula for figuring out the coordinates to answer a 
question like:

Find the image of a triangle with vertices A(0,1),B(-2,0) and 
C(-4,-5) under a rotation of 90 degrees counterclockwise about the 
origin.  

Is there a formula I can use instead of drawing a picture?

Date: 04/08/2003 at 12:42:55
From: Doctor Rick
Subject: Re: Coordinates on a grid

Hi, Reggie.

Yes, there is, but I suggest that you graph one point anyway. You can 
discover the formula by examining how the coordinates of that one 
point change under this rotation. Then you can apply the same formula 
to the other points.

Take any one of the points - I suggest C, because it's not on either 
axis - and find the coordinates of the rotated point. Hint: You can 
rotate the point by turning the whole page counterclockwise 90 
degrees. 

              y                                x
              ^                                ^
              |                                |
              |                                |
              |                                |
       -4     |                                |      -5
   -----+-----+--------->x  ===>  y<-----------+-------+----
        :     |                                |       :
        :     |                                |       :
        :     |                              -4+.......*
        *.....+-5                              |
              |                                |

Then relabel the axes the right way. You want to rotate the point but 
NOT the axes.

               y
               ^
               |
               |
               |
               |       5
   ------------+-------+---->x
               |       :
               |       :
             -4+.......*
               |
               |

What was the positive x-axis is now pointing up, so it is where the 
positive y-axis should be. Therefore, what was the x-coordinate is 
now the y-coordinate. On the other hand, what was the positive y-axis 
is now the NEGATIVE x-axis. It is pointing left. How is the new 
x-coordinate related to the old y-coordinate?

These observations will help you to write down not only the 
coordinates of the rotated point, but a rule for changing the old 
coordinates to the new coordinates. This rule could be applied to any 
point.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/