[Math] Prove that $u$ is orthogonal to $v-\operatorname{proj}u(v)$ for all vectors $u$ and $v$ in $\mathbb{R}^{n}$ where $u \neq 0$.

Question

Here's where I'm at, not sure where to go from here.

Two vectors are orthogonal if their dot product is $0$. Knowing that;

$$u \cdot (v – \operatorname{proj} u(v)) = 0$$

$$u \cdot \left(v – \frac{u \cdot v}{u \cdot u} u\right) = 0$$

$$u \cdot v – \frac{u \cdot v}{u \cdot u} u = 0$$

I'm not sure where to go from here, I know our goal is to cancel things out and I believe I need to utilize some properties of the dot product.

Answer 1

Using the notation $\;\langle u,v\rangle=u\cdot v\;$ for the inner (scalar) product of two vectors:

$$\langle u,\,v-\text{proj}_uv\rangle=\left\langle u,\,v-\frac{\langle u,v\rangle}{||u||}u\right\rangle=\langle u,v\rangle-\frac{\langle u,v\rangle}{||u|||}\langle u,u\rangle=\langle u,v\rangle-\langle u,v\rangle=0$$

because $\;||u||=\langle u,u\rangle\;$