I've got the coordinates of two points, the C in the pink square (see picture), and the coordinate of the OH. The coordinates are in 3D space; I also want to find the coordinate of CH3, but the only information I have are:
a. I know the length between C and CH3 b. I know the angle formed by CH3-C-OH (which is roughly 109 deg).
I've tried to use the Rodrigues formula to rotate the vector formed by OH-C by 109 degrees to get the coordinate of CH3, but in this method, I'm wondering what's my $k$? Would $k$ be an arbitrary axis defined by the vector defined by the C in the pink square (see above picture) and the C above that C?
If it helps, these are the coordinates:
C (above pink box) 23.916 7.640 -15.538
C (within pink box) 24.525 7.246 -16.910
CH3 23.779 6.082 -17.562
OH 24.451 8.371 -17.791 ** this is what I'd like to predict, but how would I do this?
$k$ in Rodrigues formula is a normal to plane of rotation. Since you have only 2 points, you choose this plane among all planes which contain C-CH3 bond. So, just select arbitrary point in space (say, $A$), and then two vectors C-CH3 and C-A will lie in this plain. They cross-product will be orthogonal to the plain you have chosen, and thus you can rotate around it using this formula (after you normalize it by dividing by it's length, it's exactly vector $k$).