What is the intuition behind $\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$?

Question

For any non zero vector $x$ the following inner product is zero meaning that these two matrices are orthogonal to each other. $$\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$$

What is the intuition behind this?

Answer 1

$\frac{xx^T}{\|x\|_2^2}$ is an orthogonal projection on the space spanned by $x$, while $I - \frac{xx^T}{\|x\|_2^2}$ is projection on $x$'s orthogonal complement, hence by definition/construction their inner product have to be zero.