# How to find angle after two rotations

by 1MileCrash
Tags: angle, rotations
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,339 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.