Rotating a Point 90 Degrees About the Origin

Picture the point (x, y) as an arrow from the origin. A 90 degree turn swings that arrow a quarter circle, and what was the horizontal stretch becomes the vertical stretch, and what was vertical becomes horizontal. That's why the two coordinates trade places. Only one sign flips because a quarter turn moves you into a different quadrant pattern each time: a point with two positive coordinates in the first quadrant lands in the second quadrant after a counterclockwise turn, and the second quadrant needs a negative x and a positive y. So the old y (now the x-value) must go negative, while the old x (now the y-value) stays positive. That gives the rule (x, y) → (-y, x).

Clockwise is the mirror move: turn the other way and it's the new y that flips sign instead. The rule becomes (x, y) → (y, -x). A fast way to keep these straight: in both rules the coordinates swap, and the question is just which copy gets the negative sign. Counterclockwise negates the one that becomes x; clockwise negates the one that becomes y. Test yourself with a point you can visualize, like (1, 0) sitting on the positive x-axis. Turning it 90 degrees counterclockwise should land it on the positive y-axis, at (0, 1) — and the rule (x,y) → (-y,x) gives exactly that.

This sits next to reflections and 180 degree rotations in the same toolkit of rigid motions. A 180 degree turn does negate both coordinates, since it's really two 90 degree turns stacked together — that's precisely why a single quarter turn can't do the same thing. Keep the rules for 90, 180, and 270 degree turns separate in your head, and always check your answer against a simple axis point before trusting it on a harder one.