Suppose $\alpha < \pi/2$ is an angle between vectors $\boldsymbol{u}$ and $\boldsymbol{v}$.
I am writing this way
$$tan(\alpha)$$
$$= \frac{sin(\alpha)}{cos(\alpha)}$$
$$= \frac{\frac{|\boldsymbol{u}\times\boldsymbol{v}|}{|u|\cdot|v|}}{\frac{\boldsymbol{u}\cdot\boldsymbol{v}}{|u|\cdot|v|}}$$
$$= \frac{|\boldsymbol{u}\times\boldsymbol{v}|}{\boldsymbol{u}\cdot\boldsymbol{v}}$$
Is this correct?
Now
$$(tan(\alpha))^2=$$
$$= \frac{(\boldsymbol{u}\times\boldsymbol{v})^2}{(\boldsymbol{u}\cdot\boldsymbol{v})^2}$$
$$= \frac{(u_2v_3-u_3v_2)^2 +(u_3v_1-u_1v_3)^2 + (u_1v_2-u_2v_1)^2}{(u_1v_1)^2 + (u_2v_2)^2 + (u_3v_3)^2}$$
is this also correct?
Is this really independent of vectors lengths?
I was trying to use this to perform geodesic calculations (vectors of lenth of 6 millions of meters) and was hoping it will produce smaller errors but I am obtainins something completely wrong, and only if I normalize vectors, I achieve somethong reasonable.
Where did I lost normalization or something?