Mathematically too, it turns out that the rotation matrix (which represents this process of alignment) has nine elements with six constraints imposed on them to qualify as a valid rotation matrix. The other three constraints to uniquely determine this matrix are to come from the vector pairs compared in both frames. Each of these pairs, you might think, provides three additional constraints. After all, we are to match three coordinates of both the vectors in the two frames. Remember, however, that these two vectors depict the same physical quantity in two different frames. So if two of their coordinates are matched, so is their third. This is because their magnitude (and we know this value from what quantity they represent) has to be the same in both frames. In fact, the vector pairs are all normalised to make things less complicated. <br \>

Say, A is the rotation matrix.

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The constraints then would be,

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Since a vector has 3 components, it would seem a priori that it gives 3 constraints. <br \>