I want to show the following equality:
$\vec{X}\langle\vec{Y},\vec{Z}\rangle = \langle \nabla_{\vec{X}}\vec{Y}, \vec{Z} \rangle + \langle \vec{Y},\nabla_{\vec{X}} \vec{Z} \rangle$
whereas the RHS of the equation can be expressed as:
$\langle x^j (\frac{\partial y^k}{\partial u^j} + y^i\Gamma^k_{ij})\frac{\partial\vec{F}}{\partial u^k}, \vec{Z} \rangle + \langle \vec{Y}, x^j (\frac{\partial z^k}{\partial u^j} + z^i\Gamma^k_{ij})\frac{\partial\vec{F}}{\partial u^k}\rangle $
which leaves me with a scalar, whereas the LHS gives me the product of a vector field with a scalar. Am I getting this wrong?
I appreciate any help!