Explain to me why the equivalent transformation to rotation of object clockwise 90° is equivalent to the rotation about the origin of 270° counterclockwise
The equivalent transformation to rotating an object clockwise by 90° is rotating it counterclockwise by 270° about the origin.
Let's assume we have a coordinate system with an object positioned at a certain point (x, y).
When we rotate the object clockwise by 90°, we can visualize it as a movement from its initial position to a new position. In this new position, the x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the positive of the original x-coordinate. So, if the original position is (x, y), its new position after rotating clockwise by 90° will be (-y, x).
Now, when we rotate the object counterclockwise by 270° about the origin, it can be considered as moving in the opposite direction of the previous rotation. In this case, the x-coordinate becomes the positive of the original y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. So, if the previous position was (-y, x) due to the clockwise 90° rotation, the new position after the counterclockwise 270° rotation will be (x, -y).
Hence, rotating an object clockwise by 90° is equivalent to rotating it counterclockwise by 270° about the origin.