Final answer:
To rotate a point (x, y) 90° counterclockwise about the point (1, 0), we translate the point so (1, 0) is the origin, apply the 90° rotation matrix, and then translate back, resulting in the new coordinates (-y + 1, x).
Step-by-step explanation:
To rotate a point, (x, y), 90° counterclockwise about the point (1, 0), we need to perform a two-step transformation. The first step is to translate the point so that (1, 0) becomes the origin, and the second step is to apply the 90° counterclockwise rotation matrix to the translated point. This operation is not one of the standard functions you might be used to, like a sine or cosine function, but rather it is a transformation composed of translation and rotation.
Step 1: Translate the point back so that (1, 0) is the new origin:
- (x', y') = (x - 1, y - 0) = (x - 1, y)
Step 2: Rotate the point (x', y') 90° counterclockwise about the new origin using the rotation matrix:
- The rotation matrix for 90° counterclockwise is:
[[0, -1], [1, 0]] - Apply the rotation matrix to (x', y'):
- (x'', y'') = (0·(x-1) - 1·y, 1·(x-1) + 0·y) = (-y, x - 1)
The final coordinates after the rotation will be (-y + 1, x) since we need to translate the point back to the original coordinate system.
This transformation rotates the point counterclockwise about the point (1, 0).
It's important to note that the distance of a point to the origin is invariant under rotation, meaning that rotating a point around the origin does not change its distance from the origin. This property is a fundamental aspect of rotations in the Cartesian coordinate system.