In the attached image, how do I determine where the point 'r' is in space if I know the unit vector of the axes $ \xi_1 $ and $\xi_2$ with respect to the spatial reference frame?
For example, if the unit vector of the $\xi_1$ axes is $[0, 0, 1]^T$, and the unit vector of the $\xi_2$ axes is $[-1, 0, 0]^T$, where do they intersect? I know they intersect because the cross product doesn't equal zero, but not much more than that.
For context, I am trying to find 'r' as part of the Paden Kahan subproblems used in robotics; the problem is explained here on page 119.
In addition to the two given vectors specifying the directions of the axes, you will also need one given point on the 1st axis as well as one point on the 2nd axis. Without such two points, your problem is underdetermined.
Assuming the axes do intersect and you do have the two given points in addition to the two vectors, you then can write a system of equations for the two lines in space and find the point $(x,y,z)$ that is a solution of all your equations. This solution uniquely determines the intersection point of the given two lines.