If is a rotation matrix, then This implies that and have the same entries on the main diagonal. But if is not the identity, so the rotation matrix is not completely determined by the entries on its main diagonal.
On the other hand, if is a unit vector on the axis of rotation of and if 𝜃 is the angle of rotation about that axis, then
Therefore we can express cos𝜃 in terms of , , and . Plug that value of cos𝜃 into equation (1) for each this either gives or gives two possible values of which differ only by a sign change.
We can safely assume that , because the rotation described by angle and unit vector is the same as the rotation described by angle and unit vector . That means that in general there are eight possible ways to fill in the matrix (one for each choice of the signs of each of the ), therefore eight possible solutions to the given set of equations. (For , there are four solutions if exactly one of the is zero, two solutions if two of the 𝑢𝑖 are equal to zero. There are half as many solutions if , and of course only one solution if .) Moreover, by computing the rotation matrix for the rotation by angle around the axis given by , we can compute all the unknown entries and in the rotation matrix for a specific choice of
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