- #1
ELB27
- 117
- 15
Homework Statement
Taken from "Introduction to Electrodynamics" by David J. Griffiths p.12 problem 1.8 (b):
What constraints must the elements [itex]R_{ij}[/itex] of the three-dimensional rotation matrix satisfy, in order to preserve the length of vector A (for all vectors A)?
Homework Equations
The 3D rotation matrix around an arbitrary axis:
[tex]
\begin{pmatrix}
\bar{A_x}\\
\bar{A_y}\\
\bar{A_z}\\
\end{pmatrix} = \begin{pmatrix}
R_{xx} & R_{xy} & R_{xz}\\
R_{yx} & R_{yy} & R_{yz}\\
R_{zx} & R_{zy} & R_{zz}\\
\end{pmatrix} \begin{pmatrix}
A_x\\
A_y\\
A_z\\
\end{pmatrix}
[/tex]
Thus,
[itex]\bar{A_i} = \sum_{j=1}^3 R_{ij}A_j[/itex]
where the index 1 stands for x, 2 for y and 3 for z.
The Attempt at a Solution
Since the length of the vector before and after transformation must be equal:
[itex]\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (A_i)^2[/itex]
[itex]\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (\bar{A_i})(\bar{A_i}) = \sum_{i=1}^3 \left( \sum_{j=1}^3 R_{ij}A_j \right) \left( \sum_{k=1}^3 R_{ik}A_k \right) = \sum_{i=1}^3 (A_i)^2 [/itex]
The one before the last equality I obtain from substituting the above equation for [itex]\bar{A_i}[/itex] .
I'm not sure how to proceed from here. I guess that I lack understanding of the summation notation. I tried looking it up but I still can't understand how to continue.
Any help will be greatly appreciated!