Question: HI I am wondering if there is a unique matrix that maps $(x\_1,y\_1,z\_1)$ into $(x\_2,y\_2,z\_2)$. These two vectors have equal magnitude and are defined in orthogonal 3-D basis. If there is a unique solution how can I find it by considering rotation about all three orthogonal basis?

Answer: 
There are many such matrices. If you find a single matrix $M$ which maps $v\_1=(x\_1,y\_1,z\_1)$ to $v\_2=(x\_2,y\_2,z\_2)$, then we could multiply such an $M$ by a rotation matrix whose axis goes along $v\_2$.

To get a matrix $M$ we can use $M=1/(v\_1 \* v\_1)v\_2^t v\_1$. Here by $v\_1\*v\_1$ I mean dot product of $v\_1$ with itself, and by $v\_2^t$ I mean transpose of $v\_2$, which would make it into a column vector. That way $v\_2^t v\_1$ comes out to be a matrix. You have to use $v\_1^t$ when computing the result of multiplying $M$ by $v\_1$.
In the following, c stands for cosine and s for sine (of any angle $\theta$)
$$\pmatrix{c & s\\ -s &c} \pmatrix{x\_1 \\ y\_1}$$
So in particular, we have that $s^2 + c^2 = 1$.
The matrix multiply represents the rotation about the z-axis. The magnitude of the vector is preserved as such:
$$\left|\pmatrix{x\_1 \\ y\_1}\right|^2 = x\_1^2 + y\_1^2 \quad \text{this is the original magnitude}$$
$$\left|\pmatrix{cx\_1 +sy\_1\\ -sx\_1 + cy\_1}\right|^2 = (cx\_1 +sy\_1)^2 + (-sx\_1 + cy\_1)^2$$
$$=(c^2x\_1^2+s^2y\_1^2+2csx\_1y\_1) +(s^2x\_1^2+c^2y\_1^2-2csx\_1y\_1)$$
$$=c^2(x\_1^2+y\_1^2)+s^2(x\_1^2+y\_1^2)$$
$$=(c^2+s^2)(x\_1^2+y\_1^2)$$
$$=x\_1^2+y\_1^2 \quad \text{and the magnitude is unchanged}$$

You can apply this form of matrix a few times, starting with your first point in 3-D, and reach the desired value in one coordinate at a time. So for example, to rotate about the y axis:
$$\pmatrix{c & 0& s\\0&1&0\\-s&0&c}\pmatrix{x \\ y \\ z}$$
You would only need to do this twice, as the final value would by necessity match as desired, since the magnitudes are the same for the two points.

To find c and s, use $$c=\frac{x\_1}{\sqrt{x\_1^2+y\_1^2}} \quad\text{and}\quad s=\frac{y\_1}{\sqrt{x\_1^2+y\_1^2}}$$

HINT: any two values instead of $x\_1$ and $y\_1$ in that formula gives a valid c and s for a rotation matrix.