Homework Help: Rotation Matricies

1. Oct 26, 2009

breen155

1. The problem statement, all variables and given/known data
Hey guys, I'm not sure if this bit is relevant but the first part of the question is... 'The diagram shows a triangle with vertices O, A(1,2) , B (0,2). The question I need help with is 'Each of the following matricies represents a rotation about the origin, Find the angle and direction of rotation in each case'

(1/2 -$$\sqrt{3}$$/2)
($$\sqrt{3}$$/2 1/2)

Imagine its one big set of brackets around the above matrix not 2 smaller ones :)

2. Relevant equations
I'm also not sure whether these are of relivance but x' = ax + cy and y' = bx + dy
also I have the matrix
cos$$\theta$$ -sin$$\theta$$
sin$$\theta$$ cos$$\theta$$

3. The attempt at a solution
I have been messing about with this for a while attemting to sub in co ordinates to the x' equations and y' equations but I feel I am getting nowhere. I would appreciate any help.
Breen155

2. Oct 26, 2009

aPhilosopher

$$\begin{pmatrix} a & -b \\ b & a \end{pmatrix} = \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{pmatrix}$$

$$a = \frac{1}{2}$$
$$b = \frac{\sqrt{3}}{2}$$

Can you use the definition of equality of matrices to get two equations for theta?

Last edited: Oct 26, 2009
3. Oct 26, 2009

breen155

not too sure what you mean sorry I only started teaching this to myself this morning :S erm... is it costheta - sin theta = .5 and sintheta + costheta = sqt3/2 ?

4. Oct 26, 2009

aPhilosopher

I slipped up on my notation. I fixed my last post. Try to get the equations again because they aren't right. Remember, get them by using the definition of equality for matrices.

5. Oct 26, 2009

breen155

costheta = .5 and sintheta = sqrt3/2 ? :)

6. Oct 26, 2009

aPhilosopher

correct. Now you just need to find theta.

7. Oct 26, 2009

breen155

theta is 60 degrees but how do i tell the direction of rotation, clockwish or anticlockwise ? :) (thanks for the help so far btw)

8. Oct 26, 2009

Tedjn

Remember that a positive theta corresponds, by convention, to a counterclockwise rotation. If it were a clockwise rotation of 60 degrees, then theta would be -60 degrees. This is equivalent to a counterclockwise rotation of 300 degrees, and you see that the sine and cosine of 300 degrees is exactly equal to the sine and cosine of -60 degrees. Of course, this means the question is a bit vague, since you can change the direction of rotation just by changing the angle. If you have both a direction with a correct angle, however, it should be fine.

9. Oct 26, 2009

breen155

Ok I understand now. Thanks for all the help guys