Does $(k\cdot v)k=(k\otimes k)v$, for column vectors? Is this statement correct with respect to column vectors?
$$(k\cdot v)k=k(k\cdot v)=kk^Tv=(kk^T)v=(k\otimes k)v$$
If not.. because I've been staring at the same equation for hours I suspect I've gone slightly mad, where is the error?

It's readily to show by explicitly writing down componentwise
$$lhs=\sum_j(k_jv^j)k^i$$
$$rhs=\sum_j(k^ik_j)v^j$$
Thus of course the identity holds.
