Let's talk about the difference between two spherical vectors!

Hi all,

I am writing up a paper where I have a model that includes two vectors of equal length which contain an inclination (polar angle) and a position angle (azimuthal angle). I'd like to quantify how well they are (i.e. parallel, or identical) or are not aligned (perpendicular to one another). Preferably, this quantification would result in some value between 0 (perpendicular) and 1 (parallel).

Is this something super easy? I feel like someone will respond to this question as 'Dude, you just subtract them duh', but I think it's a bit more complex than that.

Thanks for reading my post and thanks in advance for any help!

Cheers,

Travis

Re: Let's talk about the difference between two spherical vectors!

Remember that two vectors are parallel if their dot product is the product of their magnitudes (in this case, 1), and perpendicular if their dot product is zero. Just by knowing the coordinates and nothing else, you can tell whether the vectors are parallel or perpendicular.

In general, if $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$ are two unit vectors, then

$\displaystyle \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$ where $\displaystyle \theta$ is the angle between vectors a and b, $\displaystyle 0 \le \theta \le \pi$. In this case, the magnitudes of a and b are 1, so $\displaystyle \cos \theta = \vec{a} \cdot \vec{b} $.