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Chapter 3: Q17P (page 105)
Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.
The three vectors are , and .
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To see a physical example of non-commuting rotations, do the following experiment. Put a book on your desk and imagine a set of rectangular axes with the xand yaxes in the plane of the desk with the zaxis vertical. Place the book in the first quadrant with the x and yaxes along the edges of the book. Rotate the bookabout the xaxis and thenabout theaxis; note its position. Now repeat the experiment, this time rotatingabout theaxis first, and thenabout the xaxis; note the different result. Write the matrices representing therotations and multiply them in both orders. In each case, find the axis and angle of rotation.
For each of the following matrices, find its determinant to see whether it produces a rotation or a reflection. If a rotation, find the axis and angle of rotation. If a reflection, find the reflecting plane and the rotation (if any) about the normal to this plane.
The Caley-Hamilton theorem states that "A matrix satisfies its own characteristic equation." Verify this theorem for the matrix in equation (11.1). Hint: Substitute the matrixforrole="math" localid="1658822242352" in the characteristic equation (11.4) and verify that you have a correct matrix equation. Further hint: Don't do all the arithmetic. Use (11.36) to write the left side of your equation asand show that the parenthesis. Remember that, by definition, the eigenvalues satisfy the characteristic equation.
Find the characteristic frequencies and the characteristic modes of vibration for systems of masses and springs as in Figure 12.1 and Examples 3,4 , and 6 for the following arrays.
5k,m,2k,m,2k
Show that ifA and Bare matrices which don't commute, then , but if they do commute then the relation holds. Hint: Write out several terms of the infinite series for , and and, do the multiplications carefully assuming that anddon't commute. Then see what happens if they do commute
The median of a trapezoid (four-sided figure with just two parallel sides) means the line joining the midpoints of the two nonparallel sides. Prove that the median bisects both diagonals; that the median is parallel to the two parallel bases and equal to half the sum of their lengths.
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