# Angle between horizontal projections of two vectors.

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 !

• looks like curiosity, smells like homework...What have you tried? – Paul Feb 28 '17 at 16:05
• 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. – user6078387 May 8 '17 at 13:18
• Anyway, we used this as a solution: dropbox.com/s/mqkfni4916fz9gr/… There's a set of possible solutions, but we had other information that made it possible to know which solution we needed. – user6078387 May 8 '17 at 13:27

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.