Rotation of a line about a given point

Question

Suppose $A(x_1,y_1)$ and $B(x_2,y_2)$ be two points i in the Cartesian plane.If the line segment $AB$ is rotated about $B$ by angle $\theta$ in an anti clockwise direction so that $A'$ becomes $A'$,How do we find the coordinates of $A'$? Any help is duly appreciated.

Answer 1

\begin{align}A'&=B+\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}(A-B)\\&=\begin{pmatrix}x_2\\y_2\end{pmatrix}+\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x_1-x_2\\y_1-y_2\end{pmatrix}\\&=\begin{pmatrix}x_2+\cos(\theta)(x_1-x_2)+\sin(\theta)(y_1-y_2)\\y_2-\sin(\theta)(x_1-x_2)+\cos(\theta)(y_1-y_2)\end{pmatrix}.\end{align}