How to find the distance from vertical for a given 3x3 rotation matrix

Question

Given a $3 \times 3$ rotation matrix for a camera in a fixed position, how can I find the angle between where the camera is pointing and the vertical z axis?

To clarify, if the camera is imagined at the center of a globe, I am looking for the line of latitude that it would be pointing at, without caring so much about longitude (though that would be a nice bonus).

I've tried looking at conversions to Euler angles, but it seems like that is providing too much information, as rotations around each of the axis interact with each other to influence the final line of latitude - I would need a way to combine all three rotations into a final number.

Answer 1

HINT

• find the (fixed) axis of rotation by $R\vec v=\vec v$
• use $\cos \theta=\frac{\vec v\cdot \vec w}{|\vec v||\vec w|}$ to determine the angle between $\vec v$ and any vector $\vec w$