> 
> Let $v,w \in \mathbb R^n$ with $\|u\|=\|w\|=1$. Prove that there exists
>  orthogonal matrix $A$ such that $Av=w$. Also prove that $A$ can be
>  chosen such that $\det(A)=1$
> 
> 
> 

Here, $u\neq 0$ and $w\neq 0$ so I can extend these to get 

$B\_1 = \{u,x2,x3\dots x\_n\}$ and $B\_2=\{w,y2\dots y\_n\}$ both orthonormal basis of $R^n$

and define $T:R^n \to R^n $ as $T(u)=w$ and $T(x\_i)=y\_i \;\; \forall i\leq i \leq n$

here will $A=[T]\_{B\_1}^{B2}$ satisfy the given conditions?

I can't think of another way to solve this. please help