4321_HW1FA08 - A = -x ^ + 7 y ^ + 6 z ^ and B = -x ^ 6 y ^...

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Physics 4321 Homework Set 1 August 25, 2008 Due: September 3, 2008 Recall that the units vectors parallel to x, y, and z are given by x ^ , y ^ , and z ^ , respectively. 1. Is the cross product associative; i.e., is ( A × B ) × C = A × ( B × C )? If so, prove it; if not, give a counter example. 2. Find the angle between the body diagonals of a cube. 3. Use the cross product to find a unit normal n ^ to the plane that intersects the coordinate axes at the points (1,0,0), (0,2,0), and (0,0,3). 4. Evaluate (2 x ^ - 3 y ^ ) · [( x ^ + y ^ - z ^ ) × (3 x ^ - z ^ )]. 5. Calculate the scalar product, the vector product, and the angle between the vectors
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Unformatted text preview: A = -x ^ + 7 y ^ + 6 z ^ and B = -x ^ 6 y ^ + 6 z ^ . 6. Find the curl and divergence of the vector A = x x ^ xy y ^ + 3z 2 z ^ . 7. Sketch the vector field A = -y x ^ + x y ^ and calculate its curl. 8. (a) Prove that the two-dimensional rotation matrix (1.29) preserves dot products. That is, show that A y B y + A z B z = A _ y B _ y +A _ z B _ z . (b) What constraints must the elements R ij of the three-dimensional rotation matrix (1.30) satisfy in order to preserve the length of A for all vectors A ....
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This note was uploaded on 02/20/2012 for the course PHYS 1101 taught by Professor Lowellwood during the Fall '10 term at University of Houston.

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