Constraint on a three dimensional rotation to preserve vector length Given
$\begin{bmatrix}
\bar{A}^{x}\\ 
\bar{A}_{y}\\ 
\bar{A}_{z}
\end{bmatrix}
=\begin{bmatrix}
R_{xx} & R_{xy} &R_{xz} \\ 
R_{yz} &R_{yy}  &R_{yz} \\ 
R_{zx} &R_{zy}  &R_{zz} 
\end{bmatrix}
\begin{bmatrix}
A_{x}\\ 
A_{y}\\ 
A_{z}
\end{bmatrix}$
What constraints must the element $R_{ij}$ of the three dimensional rotation matrix satisfy in order to preserve the length of A for all vectors A
A crucial hint would be very helpful. No answers!
 A: The question is a bit confusing because a "rotation" matrix is one that performs rotation, and rotations don't change lengths. So, the answer to your question is that there are no constraints -- because every rotation matrix preserves lengths. Incidentally, all reflection matrices preserve lengths, too.
But, I suspect that what you really want to know is what constraints on a $3\times 3$ matrix cause it to preserve lengths.
Providing a good hint is harder than providing the answer, but here's an attempt ...
If the matrix $\mathbf{R}$ preserves lengths, then $\|\mathbf{R}\mathbf{x}\| = \|\mathbf{x}\|$ in the particular case where $\mathbf{x} = (0,0,1)$. So, specifically, when  $\mathbf{x} = (0,0,1)$, we have $\|\mathbf{R}\mathbf{x}\| = 1$, and this tells you that 
$$
R_{xx}^2 + R_{yx}^2 + R_{zx}^2  = 1
$$
If we let $\mathbf{u} = (R_{xx}, R_{yx}, R_{zx})$ denote the first column of $\mathbf{R}$, this says that $\mathbf{u} \cdot \mathbf{u} = 1$. If we let $\mathbf{v}$ and $\mathbf{w}$ denote the second and third columns, then you can probably figure out how to show that $\mathbf{v} \cdot \mathbf{v} = 1$ and $\mathbf{w} \cdot \mathbf{w} = 1$, by similar reasoning.
Next, try to figure out the values of $\mathbf{u} \cdot \mathbf{v}$ and $\mathbf{v} \cdot \mathbf{w}$ and $\mathbf{w} \cdot \mathbf{u}$.
Eventually, you'll find that the three vectors $\mathbf{u}$, $\mathbf{v}$, $\mathbf{w}$ have some very special relationship with each other, which is one way to express the constraint you're after.
Next, observe that
$$
\mathbf{R}\mathbf{R}^T =
\left| \; 
\begin{matrix}
\mathbf{u} \cdot \mathbf{u} & 
\mathbf{u} \cdot \mathbf{v} & 
\mathbf{u} \cdot \mathbf{w} \\
\mathbf{v} \cdot \mathbf{u} & 
\mathbf{v} \cdot \mathbf{v} & 
\mathbf{v} \cdot \mathbf{w} \\
\mathbf{w} \cdot \mathbf{u} & 
\mathbf{w} \cdot \mathbf{v} & 
\mathbf{w} \cdot \mathbf{w} \\
\end{matrix} \;
\right|
$$
So, once you have the values of all those dot products, you'll see that $\mathbf{R}\mathbf{R}^T = \text{<something special>}$, which is another way to state the constraint.
This is all somewhat circuitous, but that's because I'm trying to lead you in circles around the answer, rather than directly to it, since this seems to be what you want.
You can get the answer much more directly by magical manipulation of symbols, if you want: just follow the hint given in the comment by @symplectomorphic.
