Rotation of Vectors: Comparing Matrices

[ahmed markhoos]

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

Hello,

I have a problem with rotation matrices, its just a comparison problem. Θ must be 240 or -120, I don't know how the book show the answer like that, these are the two matrices

\begin{array}--1/2&\sqrt{(3)}/2\\-\sqrt{(3)}/2&-1/2\end{array}

with

\begin{array}ccosΘ&-sinΘ\\sinΘ&cosΘ\end{array}

- I tried to take element 1,1 and 2,1 , it gives 120 & -60. How is that suppose to be 240 or -120?

 

[Fredrik]

When you match the 11 and 21 components, you get two equations. Each of these equations has two solutions in the interval [0,360). Only one of the two solutions to the first equation will be a solution to the second equation as well.

 

[Ray Vickson]

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

Hello,

I have a problem with rotation matrices, its just a comparison problem. Θ must be 240 or -120, I don't know how the book show the answer like that, these are the two matrices

\begin{array}--1/2&\sqrt{(3)}/2\\-\sqrt{(3)}/2&-1/2\end{array}

with

\begin{array}ccosΘ&-sinΘ\\sinΘ&cosΘ\end{array}

- I tried to take element 1,1 and 2,1 , it gives 120 & -60. How is that suppose to be 240 or -120?

Look at the first column of your matrix: you have $\cos(\theta) = -1/2$ and $\sin(\theta) = -\sqrt{3}/2$. Since both $\sin(\theta)$ and $\cos(\theta)$ are $< 0$, in what quadrant must $\theta$ lie? Just draw a picture of you need more insight.

 

[theodoros.mihos]

-120 and 240 is the same angle.

 

[Fredrik]

Nitpick: They're equivalent, but not the same. (cos 240,sin 240) and (cos(-120),sin(-120)) are however the same point in $\mathbb R^2$.