Question

1. what is the outcome of rotating a point (2, - 3) 90 degrees counterclockwise around the point (1, 1)?

a. (4, - 1)
b. (5, 2)
c. (4, 0)
d. (0, - 2)

Explanation:

Step1: Translate points

First, translate the center of rotation $(1,1)$ and the point $(2, - 3)$ so that the center of rotation is at the origin. Subtract the coordinates of the center of rotation from the point's coordinates. Let the center of rotation $O=(1,1)$ and the point $P=(2,-3)$. The translated point $P'=(2 - 1,-3 - 1)=(1,-4)$.

Step2: Apply 90 - degree counter - clockwise rotation matrix

The rotation matrix for a 90 - degree counter - clockwise rotation about the origin is

$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

. Multiply the matrix by the column - vector of the translated point.

$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}1\\-4\end{bmatrix}=\begin{bmatrix}4\\1\end{bmatrix}$$

.

Step3: Translate back

Translate the rotated point back to the original coordinate system. Add the coordinates of the original center of rotation $(1,1)$ to the rotated and translated point. The final point is $(4 + 1,1+1)=(5,2)$.

Answer:

B. $(5,2)$