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I have a tricky question and don't know if you can solve this one:

Given:

  • I have a vector $\vec{A}$.
  • I have the angle between vector $\vec{B}$ and the projection of $\vec{B}$ on the $xz$-plane.
  • I know that $\vec{B}$ is perpendicular to $\vec{A}$.
  • I know the length of $\vec{B}$ is $1$.

How to calculate the angle between the projection of $\vec{A}$ and the projection of $\vec{B}$ in the $xz$-plane, in function of the angle between $\vec{A}$ and the $xz$-plane?

I'm curious if someone can give a decent answer, succes !

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    looks like curiosity, smells like homework...What have you tried? – 2017-02-28
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    It's not really homework, it was for a bachelor thesis assignment :p . We have to program a virtual Drone in our own 3D testbed. – 2017-05-08
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    Anyway, we used this as a solution: https://www.dropbox.com/s/mqkfni4916fz9gr/16880596_1214958358558175_1595525785_o.jpg?dl=0 There's a set of possible solutions, but we had other information that made it possible to know which solution we needed. – 2017-05-08

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I'm going to slightly change the problem: rather than have the angles & projections relative to the $xz$-plane, I'm going to do them relative to the $xy$-plane. This is really just a relabeling of the axes, but it allows us to use spherical polar coordinates for the vectors. In particular, we have $$ \vec{A} = |\vec{A}| \left[ (\sin \phi_A \cos \theta_A) \hat{x} + (\sin \phi_A \sin \theta_A) \hat{y} + (\cos \phi_A) \hat{z} \right] $$ $$ \vec{B} = (\sin \phi_B \cos \theta_B) \hat{x} + (\sin \phi_B \sin \theta_B) \hat{y} + (\cos \phi_B) \hat{z} $$ (note that $|\vec{B}| = 1$.) The angle between $\vec{A}$ and its projection on the $xy$-plane is then $\alpha_A = \frac{\pi}{2} - \phi_A$, and the angle between $\vec{B}$ and its projection is $\alpha_B = \frac{\pi}{2} - \phi_B$. WLOG we can choose the projection of $\vec{B}$ into the $xy$-plane to lie on the $x$-axis; this means that $\theta_B = 0$, and $\theta_A$ will be the angle between the vectors' projections into the $xy$-plane. Thus, our vectors are now $$ \vec{A} = |\vec{A}| \left[ (\cos \alpha_A \cos \theta_A) \hat{x} + (\cos \alpha_A \sin \theta_A) \hat{y} + (\sin \alpha_A) \hat{z} \right] $$ $$ \vec{B} = (\cos \alpha_B) \hat{x} + (\sin \alpha_B) \hat{z} $$

We also know that $\vec{A}$ and $\vec{B}$ are perpendicular, so we must have $$ 0 = \vec{A} \cdot \vec{B} = |\vec{A}| \left[ \cos \alpha_A \cos \theta_A \cos \alpha_B + \sin \alpha_A \sin \alpha_B \right] $$ which then implies that $$ \boxed{ \cos \theta_A = - \frac{ \sin \alpha_A \sin \alpha_B }{ \cos \alpha_A \cos \alpha_B} = - \tan \alpha_A \tan \alpha_B.} $$

Note that the right-hand side of this equation will not always be between -1 and 1 for certain values of $\alpha_A$ and $\alpha_B$. This is because the angles $\alpha_A$ and $\alpha_B$ cannot be freely specified and still satisfy all of above constraints, particularly the orthogonality constraint. For example, if $\alpha_A = \alpha_B = 75°$, then the angle between them cannot be greater than $30°$, and the orthogonality constraint cannot be satisfied.

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    Thanks for the answer, this was a very different method of solving the problem, but better than mine :) – 2017-02-28