I Changes under a rotation around the z-axis by an angle α

  • I
  • Thread starter Thread starter NODARman
  • Start date Start date
  • Tags Tags
    Coordinate system
NODARman
Messages
57
Reaction score
13
TL;DR Summary
.
Hi, I'm trying to solve the problem from here: https://www.physics.uoguelph.ca/chapter-1-newtonian-mechanics
Exercise 1.1: Determine how the coordinates $$x$$ and $$y$$, as well as the basis vectors $$\hat{x}$$ and $$\hat{y}$$, change under a rotation around the $$z$$ axis by an angle $$α$$. Then show mathematically that the $$r$$ vector is invariant under the transformation.

I wrote this and want to know if it's correct and how to continue:

1696531675258.png
1696531697297.png

$$r_{x y}=\sqrt{x_0^2+y_0^2}$$
$$
\sin (\beta-\alpha)=\frac{y_1}{r_{xy}}=\sin \beta \cos \alpha-\cos \beta \sin \alpha
$$
$$
\sin (\beta-\alpha)=\frac{y_1}{\sqrt{r}_{xy}}=\sin \beta \cos \alpha-\cos \beta \sin \alpha
$$
$$
\cos \beta=\frac{x_0}{r_{x y}}
$$
$$
\sin \beta=\frac{y_0}{r_{x y}}
$$
$$
\begin{array}{l}
x_1=r_{x y} \cos \alpha \cos \beta+r_{x y} \sin \alpha \sin \beta \\
y_1=r_{x y} \sin \beta \cos \alpha-r_{x y} \cos \beta \sin \alpha
\end{array}
$$
$$
\begin{array}{l}
x_1=x_0 \cos \alpha+y_0 \sin \alpha \\
y_1=y_0 \cos \alpha-x_0 \sin \alpha
\end{array}
$$
$$
z_0=z_1
$$
$$
\vec{r}_0=\left(x_0 ; y_0 ; z_0\right)
$$
$$
\vec{r}_1=(x_1 ; y_1 ; z_1) = [(x_0 \cos \alpha+y_0 \sin \alpha ) ; (y_0 \cos \alpha-x_0 \sin \alpha) ; z_0]
$$
 
Physics news on Phys.org
sorry for my bad typo
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top