# [Linear Algebra] rotational matrices

## Homework Statement

Prove Rθ+φ =Rθ+Rφ Where Rθ is equal to the 2x2 rotational matrix
[cos(θ) sin(θ),
-sin(θ) cos(θ)]

## Homework Equations

I am having a hard time trying figure our what is being asked. My question is can anyone put this into words? I am having trouble understanding what the phi may mean?

## The Attempt at a Solution

I tried to solve this graphical by plotting x and y vectors doing a rotation and plotting those and taking the product of the two rotations or x and y and finding the rotation of x+y. I don't think this is even on the right track and there must be a much simpler way to do this than graphically.

fresh_42
Mentor
A proper graphic can at least show you the way how to do it. You need only one vector at the center of rotation and then rotate it in two steps, e.g. by using a compass. Do you know what the consecutive application of two rotations mean to the two rotation matrices?

Prove Rθ+φ =Rθ+Rφ

This seems wrong. Are you sure that you copied it correctly?

A proper graphic can at least show you the way how to do it. You need only one vector at the center of rotation and then rotate it in two steps, e.g. by using a compass. Do you know what the consecutive application of two rotations mean to the two rotation matrices?
This seems wrong. Are you sure that you copied it correctly?
You are write that it isn't correct it should be Rθ+φ =Rθ*Rφ
THANK YOU!!!

fresh_42
Mentor
And have you multiplied the two?

And have you multiplied the two?
No I have not because I am unsure as to what Rφ is.

A proper graphic can at least show you the way how to do it. You need only one vector at the center of rotation and then rotate it in two steps, e.g. by using a compass. Do you know what the consecutive application of two rotations mean to the two rotation matrices?
No I am not familiar with what two applications of the rotation means.

fresh_42
Mentor
No I have not because I am unsure as to what Rφ is.
The same as with ##θ##. So ##R_φ = \begin{bmatrix}\cos φ && \sin φ \\ -\sin φ && \cos φ\end{bmatrix}##.
No I am not familiar with what two applications of the rotation means.
It means to do one rotation by ##φ## applied to some original vector ##v## and then another rotation by ##θ## applied to the resulting vector of the first.
In terms of matrices, it is - as you wrote - ## (R_θ \cdot R_φ) (v) = R_θ ( R_φ (v)) ##, i.e. you will have to multiply the two matrices.
The addition theorems for ##\cos## and ##\sin## should give you the required result.

And you can (and in my opinion should) do all this graphically, too. It will give you a nice proof of the addition theorems.