Intuition on proof of Cauchy Schwarz inequality To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind this selection. I know that if we assume this we will be able to prove the theorem, but the intuition is not clear to me.

Basically, the geometric intuition behind this proof is that the area of a parallelogram $x\wedge y$ is positive unless $x$ and $y$ are co-linear. In fact, it is precisely $||x||\cdot||y||$ when the vectors are perpendicular.
Otherwise, we can observe that the area of $x\wedge y$ is the same as of $x\wedge(y-kx)$, thus, by choosing $k$ to minimize the length of the second edge (as in the proof), we can make both edges perpendicular. The area of the resulting parallelogram must be non-negative, giving the desired inequality.
