Angle from rotation matrix

Question

I have an exam later today and I have come across something which is not covered in my lecture notes (that I know of) that I need to know the answer for!

Could someone please explain to me how you get the answer for the following question? I have been looking all over to try and solve this but not having much luck with my google searches :-) As you can tell I am a bit of a novice when it comes to Maths!

Which angle has the following rotation matrix:

     ( 0 1 )
     (-1 0 )

 [A] 0
 [B] 1/2 π
 [C] π
 [D] 3/2 π 

Thank you in advance for any help that you may be able to give me

Answer 1

The question is asking for the angle through which a vector (perpendicular to the axis of rotation) is rotated by the rotation matrix. Consider the effect of the matrix on a vector and work out how much it is rotated by. You should find the answer to be $\frac{3\pi}2$ anti-clockwise.

For more general matrices I find the best way is to look at the eigenvalues of the matrix. First consider the eigenvalues of your matrix, which one can determine by solving $$\lambda^2+1=0.$$ This gives $\lambda=\pm\mathrm i = \mathrm e^{\pm \frac{\mathrm i \pi}2}$, corresponding to a rotation of $\pm\frac\pi2,$ with the sign depending on the direction of rotation.

For a 3-dimensional rotation matrix you will find that one eigenvalue is 1. The corresponding eigenvector is the axis of the rotation.

Answer 2

To find the effect of this matrix on unit vector $\vec{I}$ multiply the matrix into $(1,0)^T$. So, $\vec{i}$ goes to $(o,-1)=-\vec{j}$. Thus a rotation of $-\pi /2$ or $3\pi /2$. Which one? Look at effect on $\vec{j}$. Multiply into $(0,1)$... You'll see that $-\pi/2$ is the right rotation whose matrix is the one given.

Answer 3

You can look at this graphically. The columns of the rotation matrix are the components of the local $\hat{i}$ and $\hat{j}$ vectors.

$$\begin{align} \hat{i} & = \pmatrix{0 \\ -1} \\ \hat{j} &= \pmatrix{1 \\ 0} \end{align}$$

Graphically you are asked to find the angle $\theta$ the produces the figure on the right: