Inner product within an inner product

Question

I have

$$\langle v-2\langle u,v\rangle u,w-2\langle u,w\rangle u\rangle$$

I tried using linearity, but I got this mess

$$\langle v,w\rangle -\langle v,2\langle u,w\rangle u\rangle -\langle 2\langle u,v\rangle u,w\rangle +\langle 2\langle u,v\rangle u,2\langle u,w\rangle u\rangle$$

I know it all cancels to $\langle v,w\rangle$ but I can't see how?

Answer 1

You ended up with $$\langle v,w\rangle -\langle v,2\langle u,w\rangle u\rangle -\langle 2\langle u,v\rangle u,w\rangle +\langle 2\langle u,v\rangle u,2\langle u,w\rangle u\rangle$$ Now use the fact you can "pull constants out of the inner product": $$\langle v,w\rangle -2\langle u,w\rangle \langle v,u\rangle -2\langle u,v\rangle\langle u,w\rangle +4\langle u,v\rangle\langle u,w\rangle\langle u, u\rangle$$ Using the symmetry $\langle u,v\rangle = \langle v, u \rangle$ and the fact $u$ is a unit vector ($\langle u,u \rangle = 1$), you end up with $$\langle v,w\rangle -2\langle u,w\rangle \langle u,v\rangle -2\langle u,v\rangle\langle u,w\rangle +4\langle u,v\rangle\langle u,w\rangle = \langle v,w \rangle$$