How do I rotate vector?

Question

It's given vector $\vec{e_{1}}=0.31\vec{e_{x}}+0.95\vec{e_{y}}$. How do I rotate that for 30 degrees counterclokwise? What I have done, I have used $x’=x\cos\theta-y\sin\theta$, $y’=x\sin\theta+y\cos\theta$, but I'm not sure if get right solution.

Answer 1

Given a point $p=(x,y)$, you can rotate it $\theta$ degrees to get a point $\tilde{p}$ by using the rotation matrix $R\in\mathbb{R}^{2\times2}$ via $$ \tilde{p} = Rp = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$ which rotates $p$ counterclockwise by $\theta$ degrees.

Notice that your results are exactly as given here :) Check out this wiki too.