Assume we have a vector $u= (u_1.u_2, u_3) \in R^3$
My problem is to find vectors $\vec w, \vec v$ such that $u= v \times w$ All vectors should be orthonormal.
If $u= (u_1, u_2, u_3)$ ,is there a way to express these vectors $\vec w, \vec v$ with respect to $\vec u$.
If there is, I would like to see an example.
I am supposed to build a matrix whose columns consist of $A =(\vec v, \vec w, \vec u)$ Remember that all vectors should be orthonormal to each other, and that $u= v \times w$
This matrix A will represent a rotation matrix, an element $\in SO(3)$
My goal is to show that $R= ASA^T$ (S a general rotation matrix around z-axis) is again a an element $\in SO(3)$, a rotation around the unit axis $u$, on the the form as you see in this article:
http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
Pick any vector $v$ orthonormal to $u$; one way to do this is to pick any vector $v'$ not parallel to $u$ and then apply the Gram-Schmidt algorithm to the pair $(u, v')$. Then, set $w := u \times v$, so that $(u, v, w)$ is an oriented orthonormal basis of $\mathbb{R}^3$; in particular, $(v, w, u)$ is also an oriented orthonormal basis of $\mathbb{R}^3$ and so $$u = v \times w$$ as desired. (If one isn't familiar with this characterization of such bases, we can alternately see this with the iterated cross product identity $$a \times (b \times c) = b \cdot (a \times c) - c \cdot (a \times b).\text{)}$$
One can show your desired result (that $R \in SO(3)$) quickly using the characterization of matrices $B \in SO(3)$ as exactly those for which ${}^t B B = I$ and $\det B > 0$.