ps7 - (b Repeat for a rotation of 45 o about the z-axis...

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PHY 504 Problem Set #7 due 22 October 2010 1. Derivation 4.14. Use these formulas to obtain expressions involving dot products for ( a × b ) × c, a × ( b × c ) , and ( a × b ) • ( c × d ) . Give three reasons why vectors do not form a group under the cross product. 2. Derivation 4.17. 3. Write down a matrix representing a rotation by 120 o about the vector (1,1,1). Do this in two ways: (a) Using the function MatrixExp in Mathematica. (b) By a geometric argument. 4. A rotation of 30 o about the x -axis is followed by a rotation of 45 o about the y -axis. (a) Find the resulting rotation matrix. About what axis is it a rotation? (Find the zero eigenvector.)
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Unformatted text preview: (b) Repeat for a rotation of 45 o about the z-axis followed by a rotation of 30 o about the x-axis. 5. Show that exp( ε A )exp( B ) = exp( A + B + 2 2 [ A , B ] + O ( 3 )) where A and B are matrices and is small. 6. The square has an eight-element symmetry group of rotations and reflections. (a) Write down a set of 2 × 2 matrices representing this group. Is this group commutative (Abelian)? (b) Give names to the 8 elements and construct their multiplication table. Verify that the product of any two reflections is a rotation. (c) How many elements are in the symmetry group of the cube?...
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