Find the matrix A of the linear transformation T from R2 to R2 that rotates any vector through an angle of 30∘ in the clockwise direction.
Heres what I did so far :
I plugged in 30 into the general matrix
\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}
which turns into this:
\begin{bmatrix}\cos 30&-\sin 30\\\sin 30 &\cos 30 \\\end{bmatrix}
resulting in:
\begin{bmatrix}\sqrt{\frac{3}{2}}&-\frac{1}{2}\\ \frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}
now, since a clockwise rotation is: \begin{bmatrix}0&1\\ -1&0\end{bmatrix}, I set up a matrix multiplication like this and multiplied the two matrices together which ends up with this:
\begin{bmatrix}\frac{1}{2}&\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}:
However I got it wrong, can anyone help?
Best Answer
Hint.
You should not multiply the two matrices together. Your result is essentially firstly rotating counterclockwise by $\pi/6$, then clockwise by $\pi/2$; it ends up with rotating counterclockwise through the angle $\pi/6-\pi/2$.
Instead, note that rotating clockwise $\theta$ is the same as rotating counterclockwise $2\pi-\theta$.
Your "general matrix" is rotating counterclockwise by $\theta$.