Inner Products why is $\langle \langle x, v_i \rangle v_i, y \rangle=\langle x,v_i \rangle\langle v_i,y \rangle$

Question

I have seen in a few different proofs $\langle \langle x, v_i \rangle v_i, y \rangle=\langle x,v_i \rangle\langle v_i,y \rangle$ by linearity but I can't seem to wrap my head around why. Could someone please explain this

Answer 1

$\langle x, v_i \rangle$ is a scalar, so it just comes from the linearity axiom of $\langle a\vec v, \vec w \rangle=a\langle \vec v, \vec w \rangle$

Answer 2

Since $\lambda=\langle x, v_i\rangle$ is a scalar. then $$ \langle\langle x,v_i\rangle v_i,y\rangle=\langle\lambda v_i,y\rangle=\lambda\langle v_i,y\rangle=\langle x,v_i\rangle\langle v_i,y\rangle. $$