(*^

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Rotation matrices (3 x 3)
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This notebook concerns the 3 x 3 rotation matrix shown at the top of p. 191 of the text.
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Since (a,b,c) is a unit vector, we know that a^2 + b^2 + c^2 = 1.   We will tell
Mathematica this  by creating a rule  (associated to the variable c) which replaces c^2 by 1-a^2-b^2, whenever simplification occurs.
;[s]
3:0,0;81,1;92,2;215,-1;
3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;
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c/: c^2 = 1 - a^2 - b^2
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Now define the rotation matrix:
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R = {{a^2 (1 - Cos[t]) + Cos[t],
      a b (1 - Cos[t]) - c Sin[t],
      a c (1 - Cos[t]) + b Sin[t]},
     {a b (1 - Cos[t]) + c Sin[t],
      b^2 (1 - Cos[t]) + Cos[t],
      b c (1 - Cos[t]) - a Sin[t]},
     {a c (1 - Cos[t]) - b Sin[t],
      b c (1 - Cos[t]) + a Sin[t],
      c^2 (1 - Cos[t]) + Cos[t]}};
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Show that this matrix is orthogonal, that it has determinant 1, and that (a,b,c) is an eigenvector of it, corresponding to the eigenvalue 1.  Use
Simplify and Expand as needed.
^*)