Binomial theorem for vectors

Question

Let $u, v$ be vectors in n-dimensional Euclidean space. Then does the following hold? :

$|u+v|^p=|u|^p+p|u|^{p-2}u \cdot v +\dfrac {p(p-1)}2|u|^{p-3}|v|u \cdot v+\dots+|v|^p$

Answer 1

No. Try $n=1, u = (1,0), v=(-1,0)$. Die LHS is $0$, while the RHS is $1+1=2$.