How to rotate a point on a Cartesian plane around something other than the origin (0,0)

Question

What I understand: Rotating a point around the origin using rotation rules. All Rotation Rules

What I don't understand: I tried searching how to rotate other than the origin but I don't get how they make the point equivalent to the custom rotation point(other than the origin like 1,0)

What I am looking for: A step by step example on how to rotate around something other than the origin. I am not in highschool yet so please give me a easy/understandable answer. I have an example question attached. Just open the images. One has the question, other has the point with grid.

Answer 1

If you want to rotate a point $P$ about a point $A$ other than the origin, then calculate the difference vector $(P - A)$ and rotate that as you would usually, and finally add the vector $A$ to the resultant vector of rotation.

Example: Suppose I want to rotate $P = (5, 2) $ about the point $A = (3, 1)$ by $90^\circ$ anti-clockwise. First, we compute $(P - A)$

$V = P - A = (5, 2) - (3, 1) = (2, 1) $

Next, we rotate $V$ by $90^\circ$, this will give

$V' = R_{90^\circ} V = (-1, 2) $

And finally, we'll add $A$ to $V'$ to obtain the image $P'$

$P' = A + V' = (3, 1) + (-1, 2) = (2, 3) $

Answer 2

The general transformation you're looking for is $$\left( \begin{array}{c} x_1\\ y_1 \end{array}\right) = \left( \begin{array}{cc} \cos \theta & -\sin \theta\\ \sin\theta & \cos\theta \end{array}\right) \left( \begin{array}{c} x_0-x_c\\ y_0-y_c \end{array}\right) + \left( \begin{array}{c} x_c\\ y_c \end{array}\right) $$ Where the point $(x_0,y_0)$ gets rotated into $(x_1,y_1)$ by angle $\theta$ around point $(x_c, y_c)$.