Introduction
This post is to aid students in visualizing the connection between the geometry of rotating points about the origin and how this connects to the algebraic facts (The Rules) that are expected to be memorized for many tests.
(Wooooooo……inverted colors!!! You can change this in the settings using the wrench icon in the top right of the Desmos screen. Explore the other things you can change, like grid, arrows, steps, etc.)
The Rules
Remember that a positive rotation about the origin is in the counterclockwise direction.
Here are the facts:
Switch/FlipFirst!
A helpful way to remember these rules?
Switch/FlipFirst!
You start at some coordinate (x,y). Each positive 90° increase in the angle of rotation has the same pattern with the coordinates: You SWITCH the coordinates; then you FLIP the FIRST coordinate’s sign.
For example, start with point (3,-5).
- To rotate 90°, SWITCH the coordinates [to (-5,3)] and then FLIP the new FIRST coordinate’s sign [to (+5, 3)], or simply (5,3).
- To rotate 180°, SWITCH the previous coordinates (after the 90° rotation) again [to (3,5)] and then FLIP the new FIRST coordinate’s sign to (-3,5).
- Continue in this way for 270° and 360°. If you arrive back where you started at 360°, this is a ‘positive sign’ you did this correctly. (Answers at the bottom of this section.)
For example, start with point (3,-5).
To rotate in 90° increments in the negative direction, you do the reverse: that is the FlipFirst/Switch method.
For example, start with point (3,-5).
- To rotate – 90°, FLIP the first coordinate sign [to (-3,-5)] and then SWITCH the coordinates [to (-5,-3)], or simply (5,3).
- And continue on your merry way counterclockwise about the origin…
With this method, you no longer have to simply memorize the rules. You now have a simple process you can use anytime. BRILLIANT RIGHT? I know, I know. I came up with it myself…..and I’m sure no one else in the history of mathematics has ever thought of it ever. (Leave comments / abundant praise / links to high value online gifts in the Response Section below.)
Answers: 270°→(-5,-3), 360°→(3,-5)
The Geometry
The Start
Notice the geometric relationship of the top right point to the origin. To get from the origin to the Original Position of the point, you must travel to the right “orange” and up “blue”, or right 4 and up 2.
Keep in mind right and up are considered positive, so this point is positive in both components.
After 90° Rotation
Now observe the geometric relationship of the origin to the point After Rotation: Now you must travel LEFT blue and UP orange. So the blue component is now negative in the horizontal direction, whereas the orange component remains positive but is now in the vertical direction.
Connecting the Geometry to the Algebraic Coordinates
As is indicated in the image, for the original position of the point, the orange length is x, and the blue length is y.
When the point is rotated, the side lengths of the rectangle formed with the axes stay the same. It is simply their direction that changes.
- So what was an right-orange-length-‘x’-movement in the positive x direction becomes an up-orange-length-‘x’-movement, which is in the positive y direction. Combining these aspects give us the y-coordinate of the new point, x.
- And what was an up-blue-length-‘y’-movement in the positive y direction now becomes left-blue-length-‘y’-movement, which is in the negative x direction. Combining these aspects gives us the x-coordinate of the new point, -y.
Combined, for a 90° rotation, the coordinate (x,y) goes to (-y,x), which is often written as: (x,y)→(-y,x).
Still confused? I paid extra care to do so…..The Desmos Demo can illustrate this concept more simply. Keep your eye on the labels of the rectangles.