I have a question about performing rotation in the topic geometry, How can you find the coordinates when X,Y, is rotated counterclockwise 90 degrees

Question

How can you find the coordinates when x,y is rotated counterclockwise 90 degrees. I looked up all the steps too performing rotation but still don't get it.

Answer 1

Well, in order to "get it," you need an explanation.  And, an explanation with a picture really, really helps (most people are good visual learners).  PLZ draw onel

 

The Cartesian Coordinate System starts with x-y coordinates (but also has z for 3-D).  The positive x-coordinates are to the right (like East direction) and the positive y-coordinates are up (like North) in a plane (note: z comes off the page).  Also, counterclockwise is the opposite direction of an analog (not digital) clock (the old-fashioned kind, since everyone uses a cell phone rather than a watch anymore).  90° is a "right angle."

 

So, to rotate the axes (not the point) 90° counterclockwise, that makes "North" the new x-axis and it makes "West" the new y-axis.

 

The new x-coordinate (start with x because it may be easier) takes the old value of y.  In English, however north or south this point used to be, that is now the value of x.

 

The new y-coordinate is a little trickier (I think).  The new positive y is what used to be negative x. So, however west or east (not east or west) this point used to be is now the value of y.

 

So, this means that a point (a,b) [that is, x=a and y=b) is transformed into (b,-a) [that is, x=b and y=-a)  [note: I did not use (x,y) because I think that is confusing.]

Answer 2

I do not know if you have a background in matrix algebra, but if you do, here is the way to accomplish a rotation in the x-y plane:

 

Suppose you rotate by an angle θ from the x-axis toward the y-axis.  Then the y-axis also gets rotated by the same angle θ toward the negative x-axis.  The new coordinates are (x',y').  You can draw a diagram to see that

 

x' = x cosθ + y sin θ

y' = -x sin θ + y cos θ

 

You can represent this as the 2x2 matrix whose first row is (cos θ, sin θ) and second row is (-sin θ, cos θ), multiplying the column vector (x,y), resulting in (x',y').  Replace θ with -θ to rotate in the opposite direction.

 

This is making use of the Euler rotation matrices, also called direction cosine matrices.

 

Note this gives the same result as in David's answer for the 90º counterclockwise rotation.  He started with (a,b) and got (b,-a).  For θ = 90º, the Euler matrix rows become (0,1) and (-1,0).  Multiply a column vector (a,b) by that matrix and the result is (b,-a).

Answer 3

To accomplish this rotation:

 

(x, y) ⇒ (y, -x)

 

For example

 

(2, 3) ⇒ (3, -2)

Answer 4

Actually, I believe the rotation for 90 degrees counterclockwise is:

 

(x, y) -> (-y, x)

 

So, (2, 3) -> (-3, 2)