Direction vector from 2 orthogonal angles I have an acoustic sensor that measures angle on a single axis. If the sensor is pointing upwards, then I can measure an angle $\theta$ that has the range $-\pi/2 < \theta < \pi/2$. A value of $\theta = 0$ means that an object (sound source) is directly above it. In other words, it measures the angle from a surface normal with respect to the ground, on a single axis.
If I mount a second sensor with a baseline sitting at 90 degrees from the first sensor, rotated on the z axis, I measure two angles, $\theta$ and $\phi$. $\theta$ defines a unit vector on the x,z plane (z is up) and $\phi$ defines a unit vector on the y,z plane. Both vectors point upwards when the object is directly above the sensor. $\theta$ varies as the object moves in the $\pm$ x direction and similarly $\phi$ varies according to the position on the y axis. It cannot measure distance, only direction.
How can I turn these two measurements ($\theta$ and $\phi$) into a direction vector that points towards the object? Note that this is not the same as the spherical coordinate system that measures azimuth and inclination. In my case $\theta$ and $\phi$ represent angles from a surface normal on 2 orthogonal axes.



Does this figure help you? The "perspective of sensor" triangles give you the angles you need.
EDIT:
The boxed equations are the ones you want, if i've interpreted your question correctly. (See new image)
You can figure out a direction vector, because my $\beta$ and $\alpha$ are the co-latitude and azimuthal angles in polar coordinates, respectively.

