# How to find angle after two rotations

1. Jan 23, 2013

### 1MileCrash

I have coordinate system A with bases a, b, c.

Say I rotate the whole system 30 degrees, so that the angle between a and a' is 30 degrees.

Then I make another rotation so that this plane of rotation is perpendicular to that of the old one.

What is the angle between a and a' now?

I am trying to find the angles to use in a tensor transformation law, but I am having problems understanding what the angles will be between the old and new axes when a transformation isn't just a single rotation in one plane of the system.

Tia

2. Jan 24, 2013

### HallsofIvy

You can write any rotation as a matrix multiplication. Then two rotations is given by the product of the two matrices.

For example, if you wrote 30 degrees around the z- axis, the rotation is given by
$$\begin{bmatrix} cos(30) & -sin(30) & 0 \\ sin(30) & cos(30) & 0 \\ 0 & 0 & 1\end{bmatrix}= \begin{bmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ \frac{1}{2} & \frac{\sqrt{3}}{2}& 0 \\ 0 & 0 & 1\end{bmatrix}$$

A rotation around the y-axis, through 30 degrees is given by
$$\begin{bmatrix} cos(30) & 0 &-sin(30)\\ 0 & 1 & 0 \\ sin(30) & 0 & cos(30) \end{bmatrix}= \begin{bmatrix}\frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2}\end{bmatrix}$$

The two rotations together would be given by the product of the two matrices.

3. Jan 24, 2013