The equations you have given are for a rigid body. If you are comfortable regarding the earth as a rigid body within a zone of interest*, then you need to measure the x,y,z displacements at three non-collinear points in order to compute (or estimate) the rotation. You can understand this by considering a cube. If I know the x,y,z displacement of one corner of the cube during a movement, I will not be able to determine if the cube rotated, or translated, or did a little of both. But if I know the x,y,z displacements of three corners of the cube, then I will be able to determine how much translation and rotation occurred. If I know the x,y,z displacements of four or more corners, then I can use the "extra" information to estimate whether the cube is in fact moving as a rigid body, or not.
A mathematical way of understanding the requirement for displacements at three points is that if I only have 
at one point, I will not be able to compute 
and the other partials. I need 
at three non-collinear points in order to estimate the different partials.
at one point, I will not be able to compute and the other partials. I need at three non-collinear points in order to estimate the different partials. * In seismology, one often does NOT regard the earth as a rigid body.
