Show the mathematical work for rotating an origin 270 degrees

Answer 1

To rotate a point (x, y) by 270 degrees counterclockwise around the origin, you can use the following rotation matrix:

|cos(theta) -sin(theta)| * |x| = |x'|
|sin(theta) cos(theta)| |y| |y'|

For θ = 270 degrees (or 3π/2 radians), substituting the values, the rotation matrix becomes:

|cos(3π/2) -sin(3π/2)| * |x| = |x'|
|sin(3π/2) cos(3π/2)| |y| |y'|

Simplifying using the trigonometric values, we have:

|0 -1| * |x| = |x'|
|1 0| |y| |y'|

Expanding the matrix multiplication further:

0*x - 1*y = x'
1*x + 0*y = y'

Simplifying these equations, we get:

- y = x'
x = y'

Therefore, to rotate a point (x, y) by 270 degrees counterclockwise around the origin, the new coordinates (x', y') can be determined by swapping the x and y values and changing the sign of the new y value:

x' = -y
y' = x