I asked this [on MSE](https://math.stackexchange.com/questions/471675/does-this-cross-product-norm-inequality-hold) over a month ago, but the one answer I got doesn't seem to work.
  
  
  
  
  

Let $\times$ denote the cross-product. $\;$ Is it the case that
  




> 
> For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in $\mathbf{R}^3$, $\;\;\;\; \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{z}\hspace{.02 in}\right|\right| \:\: \leq \:\: \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{y}\hspace{.02 in}\right|\right| \hspace{.02 in}+\hspace{.02 in}\left|\left|\hspace{.03 in}\mathbf{y} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{z}\hspace{.02 in}\right|\right| \;\;\;\;\;$.
> 
> 
> 


?
  
  
  

(If yes, then $\;\;\; \langle [\mathbf{x}]\hspace{.01 in},\hspace{-0.03 in}[\hspace{.02 in}\mathbf{y}] \rangle \: \mapsto \: \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{y}\hspace{.02 in}\right|\right| \;\;\;$ defines a nice metric on the projective plane.)
  
  




Does this cross-product norm inequality hold?
The alleged inequality is true. 


Using the definition of the [(vector) cross product](http://en.wikipedia.org/wiki/Cross_product), for unit vectors $x, y \in R^3$ the original claim boils down to showing that 
\begin{equation\*}
 d(x,y) = \sin(\cos^{-1}(x^Ty))
\end{equation\*}
is a distance. But it turns out that $d(\cdot,\cdot)$ is actually is a distance on unit vectors in $R^n$.


Showing the above function to be a distance is equivalent to showing that
\begin{equation\*}
 \sqrt{1 - (x^Ty)^2}
\end{equation\*}
is a distance over the set of unit vectors in $R^n$. But this is a known result of [Wang and Zhang](http://www.jstor.org/stable/2974909), *"A trace inequality for unitary matrices"*, AMM, 1994, pp. 453-455.


*EDIT:* Also worth comparing with the recent determinant based distance described [in this MO question](https://mathoverflow.net/questions/141483/is-this-a-metric-on-the-grassmannian-manifold/141535#141535) (which suggests that actually after suitable normalization, other symmetric matrix functions should also generate distances, e.g., $d(X,Y) := [1-\text{trace}(\wedge^k(X^TY))]^{1/2}$.


