Question

rotations clockwise about the origin
rotations of 90° rotations of 180° rotations of 270°
(x,y)→( _, _) (x,y)=( _, _) (x,y)→( _, _)
rotations counterclockwise about the origin
rotations of 90° rotations of 180° rotations of 270°
(x,y)→( _, _) (x,y)=( _, _) (x,y)→( _, _)

Explanation:

Step1: Recall rotation rules

For a rotation of 90 - degrees clockwise about the origin, the rule is \((x,y)\to(y, - x)\).

Step2: For 180 - degrees clockwise

The rule for a 180 - degrees rotation (clockwise or counter - clockwise) about the origin is \((x,y)\to(-x,-y)\).

Step3: For 270 - degrees clockwise

The rule for a 270 - degrees clockwise rotation about the origin is \((x,y)\to(-y,x)\).

Step4: For 90 - degrees counter - clockwise

The rule for a 90 - degrees counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\).

Step5: For 180 - degrees counter - clockwise

The rule for a 180 - degrees counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\).

Step6: For 270 - degrees counter - clockwise

The rule for a 270 - degrees counter - clockwise rotation about the origin is \((x,y)\to(y, - x)\).

Answer:

Rotations clockwise about the origin:
Rotations of 90°: \((x,y)\to(y, - x)\)
Rotations of 180°: \((x,y)\to(-x,-y)\)
Rotations of 270°: \((x,y)\to(-y,x)\)
Rotations counter - clockwise about the origin:
Rotations of 90°: \((x,y)\to(-y,x)\)
Rotations of 180°: \((x,y)\to(-x,-y)\)
Rotations of 270°: \((x,y)\to(y, - x)\)