Here's a question that appears on my homework (shown in the image above). I'm stuck on how to prove part a). I don't know what to make of the highlighted part. Inner product division and multiplication??
$$\langle v-p,w\rangle\\ = \left\langle v-\left(\frac{\langle v,w\rangle}{\langle w,w\rangle}w\right),w\right\rangle \ \ \ \ \ \text{By substitution of p}\\ = \langle v,w\rangle-\left\langle\left(\frac{\langle v,w\rangle}{\langle w,w\rangle}w\right),w\right\rangle \ \ \ \ \ \text{By property (iii)}\\ \text{...}\\ =0$$
After that, I don't know what to do about the division and multiplication of the inner product while still keeping it general and using variables.
Edit: And I know you can multiply w through via property (iv), but I don't know what good that will do.
After posting this question, I linked it to someone else and they explained it to me. I believe this is the correct answer and I'll lay it out here for anyone else:
$$\langle v-p,w\rangle\\ = \left\langle v-\left(\frac{\langle v,w\rangle}{\langle w,w\rangle}w\right),w\right\rangle\\ = \langle v,w\rangle-\left\langle\left(\frac{\langle v,w\rangle}{\langle w,w\rangle}w\right),w\right\rangle\\ = <v,w>-(\frac{<v,w>}{<w,w>})<w,w>\\ = <v,w>-(\frac{<v,w><w,w>}{<w,w>})\\ = <v,w>-<v,w>\\ = 0$$