I am working on a crystallographic problem. I am creating a 2D cross-section or slice of a surface in particular crystallographic directions. (I have found the angles of the extrema with respect to phi and theta and am now trying to plot them; since my equation is a slice of a longitudinal surface, I am holding phi = 45 degrees).
My question: how do you find the components of another vector, like [111], given the angle between the z-axis (theta = 54.7 degrees) and the components that correspond to the z-axis ([001], known as x3? Once I find mathematically that the middle direction is [111], I can use the same process to find that the x-axis direction is [110], since phi is known to be 45 degrees. This question is similar to this video (https://www.youtube.com/watch?v=0g3CQEEjbH8), except in the video, the components of both vectors are known and the angle between them is unknown; in my question, one vector and one angle are known and the other vector is unknown.
I've attached a few pictures of what I'm trying to accomplish.
2D representation 3D representation
Please let me know if you need more background, but I'm trying to avoid distracting from my current question. (I am comfortable with everything up until this point.)
Thank you in advance for your help!