Constraints the elements of the 3D-rotation matrix must satisfy

[ELB27]

Homework Statement

Taken from "Introduction to Electrodynamics" by David J. Griffiths p.12 problem 1.8 (b):
What constraints must the elements R_{ij} 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:
<br /> \begin{pmatrix}<br /> \bar{A_x}\\<br /> \bar{A_y}\\<br /> \bar{A_z}\\<br /> \end{pmatrix} = \begin{pmatrix}<br /> R_{xx} &amp; R_{xy} &amp; R_{xz}\\<br /> R_{yx} &amp; R_{yy} &amp; R_{yz}\\<br /> R_{zx} &amp; R_{zy} &amp; R_{zz}\\<br /> \end{pmatrix} \begin{pmatrix}<br /> A_x\\<br /> A_y\\<br /> A_z\\<br /> \end{pmatrix}<br />
Thus,
\bar{A_i} = \sum_{j=1}^3 R_{ij}A_j
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:
\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (A_i)^2
\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
The one before the last equality I obtain from substituting the above equation for \bar{A_i} .
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!

 

[Ray Vickson]

Homework Statement

Taken from "Introduction to Electrodynamics" by David J. Griffiths p.12 problem 1.8 (b):
What constraints must the elements R_{ij} 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:
<br /> \begin{pmatrix}<br /> \bar{A_x}\\<br /> \bar{A_y}\\<br /> \bar{A_z}\\<br /> \end{pmatrix} = \begin{pmatrix}<br /> R_{xx} &amp; R_{xy} &amp; R_{xz}\\<br /> R_{yx} &amp; R_{yy} &amp; R_{yz}\\<br /> R_{zx} &amp; R_{zy} &amp; R_{zz}\\<br /> \end{pmatrix} \begin{pmatrix}<br /> A_x\\<br /> A_y\\<br /> A_z\\<br /> \end{pmatrix}<br />
Thus,
\bar{A_i} = \sum_{j=1}^3 R_{ij}A_j
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:
\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (A_i)^2
\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
The one before the last equality I obtain from substituting the above equation for \bar{A_i} .
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!

Google "orthogonal matrix"

 

[vela]

The Attempt at a Solution

Since the length of the vector before and after transformation must be equal:
\sum_{i=1}^3 (\bar{A_i})^2 = \sum_{i=1}^3 (A_i)^2
\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
The one before the last equality I obtain from substituting the above equation for \bar{A_i} .
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.

First thing you should do is learn Einstein's summation convention (Einstein's greatest contribution to physics): repeated indices imply summation. It'll save you a lot of tedious writing. Using this convention, you have
$$\bar{A}_i \bar{A}_i = (R_{ij}A_j)(R_{ik}A_k).$$ No annoying sigmas to keep writing.

The relationship has to hold for all vectors, so try using $A = (1, 0, 0)$, for example, to show that the first column of $R$ has to be a unit vector. For other choices of $A$, you can show that different columns are orthogonal to each other.

 

[ELB27]

Google "orthogonal matrix"

Thanks, I didn't know about these matrices!

First thing you should do is learn Einstein's summation convention (Einstein's greatest contribution to physics): repeated indices imply summation. It'll save you a lot of tedious writing. Using this convention, you have
$$\bar{A}_i \bar{A}_i = (R_{ij}A_j)(R_{ik}A_k).$$ No annoying sigmas to keep writing.

The relationship has to hold for all vectors, so try using $A = (1, 0, 0)$, for example, to show that the first column of $R$ has to be a unit vector. For other choices of $A$, you can show that different columns are orthogonal to each other.

Thank you! So if I understand correctly, I just pick 2 convenient vectors like $\vec{A} = <1,0,0>$ and $\vec{B} = <1,1,0>$ and generalize the results I get for all choices? Am I allowed to do it without proving these relations in general?

EDIT: After some more readings I think I finally get it. Thanks again.