# Rotating Vector

1. Mar 5, 2007

### yoyo

1. The problem statement, all variables and given/known data
A vector x in R^2 is rotate twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify that these two expressions are equivalent

2. Relevant equations
rotation matrix R=[cos, -sin; sin, cos]

3. The attempt at a solution

I can only think of one expression:

(R)(R)(x).

Could (R^2)(x) be the other one? How would i prove that this is equivalent?

Last edited: Mar 5, 2007
2. Mar 5, 2007

### ZioX

How about just rotating once by 2theta?

3. Mar 5, 2007

### yoyo

follow up question: a vector x in R^2 is rotated n times through an angle theta. Find two expressions for the matrix representing this rotation. what identity is implied.

if what Ziox said is true then it should be [cos (theta)n, -sin(theta)n; sin (theta)n, cos(theta)n]

but i dont see what identity this implies?

4. Mar 6, 2007

### cristo

Staff Emeritus
You should probably find a second expression for the matrix first.

5. Mar 6, 2007

### HallsofIvy

Staff Emeritus
First, what is
$$\left(\begin{array}{cc}cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta)\end{array}\right)^2$$?

Second, can you use trig identities to write that in terms of $cos(2\theta)$ and $sin(2\theta)$?