math for physics/vector calculus

Vector Calculus (3) - Properties of Rotation Matrices

이너피스! 2021. 7. 14. 16:40

6. Properties of Rotation Matrices

$ (\alpha ,\beta ,\gamma )\rightarrow (\alpha ',\beta ',\gamma ') $

$ (x_{1},x_{2},x_{3}) $ coordinate에서 $ \Theta $ 만큼 회전한 $ (x_{1}',x_{2}',x_{3}') $ coordinate

 

1) $ \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =1 $

$ x_{1}=r\cos \alpha , x_{2}=r\cos \beta , x_{3}=r\cos \gamma $

$ r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}\cos ^{2}\alpha +r^{2}\cos ^{2}\beta +r^{2}\cos ^{2}\gamma $

$ \therefore \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma=1 $

 

2) $ \cos \theta =\cos \alpha \cos \alpha '+\cos \beta  \cos \beta \alpha '+\cos \gamma  \cos \gamma ' $

$ x_{1}'=r'\cos \alpha' , x_{2}'=r'\cos \beta' , x_{3}'=r'\cos \gamma' $

$ r^{2}+r'^{2}-2rr'\cos \Theta = (x_{1}-x_{1}')^{2}+(x_{2}-x_{2}')^{2}+(x_{3}-x_{3}')^{2} $

$ rr'\cos \Theta = x_{1}x_{1}'+x_{2}x_{2}'+x_{3}x_{3}' $

$ \therefore \cos \theta =\cos \alpha \cos \alpha '+\cos \beta  \cos \beta '+\cos \gamma  \cos \gamma ' $

 

 

7. Kronecker Delta

 

$ (\lambda _{11},\lambda _{12},\lambda _{13}) $ : direction cosines of the $ x_{1}' $-axis in the $ (x_{1},x_{2},x_{3}) $ system

$ (\lambda _{21},\lambda _{22},\lambda _{23}) $ : direction cosines of the $ x_{2}' $-axis in the $ (x_{1},x_{2},x_{3}) $ system

$ \lambda _{11}\lambda _{21}+\lambda _{12}\lambda _{22}+\lambda _{13}\lambda _{23}=\cos \Theta =\cos \frac{\pi }{2}=0 $

$ \Rightarrow \sum_{j}^{}\lambda _{ij}\lambda _{kj}=0 $, if $ i\neq k $

 

$ \lambda_{11}^{2}+\lambda_{12}^{2}+\lambda_{13}^{2}=\cos \Theta =\cos 0=1 $

$ \Rightarrow \sum_{j}^{}\lambda _{ij}\lambda _{kj}=1 $, if $ i=k $

 

Kronecker Delta Symbol: $ \delta _{ik} $

$ \sum_{j}^{}\lambda _{ij}\lambda _{kj}=\delta _{ik} $: Orthogonality Condition

$ \delta _{ik}=\left\{\begin{matrix}
0 (i\neq k) \\ 
1 (i= k)
\end{matrix}\right. $