Let "$\cdot$" the dot product in $\mathbb{R}^{3}$, and $x,y,z\in \mathbb{R}^{3}$ with $\left\|x\right\|=\left\|y\right\|=\left\|z\right\|=1$. Show that
$$\left|z\cdot y -(x\cdot y)(z\cdot x)\right|^{2}\leq \left(1-\left|x\cdot y\right|^{2}\right)\left(1-\left|z\cdot x\right|^{2}\right).$$
Remark: I tried to expand the first equation considering the vectors as triples but it is too cumbersome, I think that this should be able to be demonstrated using linear algebra tools, but I have not been able to identify which ones are useful.
Also, note that $x\cdot y $ turns out to be the norm of the projection of $ x $ into $ y $.