Let $W$ be a vector field of constant length and let $v$ be a vector. Show that the covariant derivative, $\nabla_vW$ and $W$ are orthogonal I know that the covariant derivative of $W$
 with respect to $v$ is the tangent vector
$\nabla_vW=W(p+tv)′(0)$
at the point $p$
We want to show that the dot product of $W$ and 
$\nabla_vW$ is zero which would imply orthogonality 
So if $||W||=L$, $L$ is constant, would it be true that our vector $v=(a,b,c)$, is a vector of just coefficients without any variable $x,y,z$?
So any $v$ in the tangent space is constant?
and...
I want to write a clear proof of the above claim, but am not sure where to go from here

You need Koszul's formula. It tells you that: 
$\langle\nabla_XY,Z\rangle = \frac{1}{2}(X\langle Y,Z\rangle+Y\langle X,Z\rangle-Z\langle X,Y\rangle+\langle[X,Y],Z\rangle-\langle[X,Z],Y\rangle-\langle[Y,Z],X\rangle)$
Then we substitute $X=V$, $Y=Z=W$ and we use the fact that $ V\langle W,W\rangle=0$ since $W$ has a constant norm and it's derivative by any vector field (in this case $V$) is zero. Explicitly, we have the following:
$\langle\nabla_V W,W\rangle = \frac{1}{2}(V\langle W,W\rangle+W\langle V,W\rangle-W\langle V,W\rangle+\langle[V,W],W\rangle-\langle[V,W],W\rangle-\langle[W,W],V\rangle)=0$
