Mathematics 138
Problem Set 3 due Wednesday, March 2, 2005
For purposes of this problem set, 53o means arctan 4/3, so that sin 53o = 0.8  and cos 53o = 0.6
74o means arctan 24/7, so that sin 74o = 0.96 and cos 74o = 0.28.
If you know how to use Mathematica (free download for Harvard undergraduates) to do matrix calculations, you may do so and just attach a printout.
Remember that by convention "AB" means "B followed by A".
1. Ryan, problem 8 on p. 35.
2. Find a 3x3 matrix that represents a counterclockwise rotation through 53 degrees, followed by a translation of 6 units along the positive x axis, followed by a translation of 2 units along the positive y axis.  Determine what point in the plane is held fixed by this overall transformation.
3. Find a 3x3 matrix A that represents reflection in the line 2x + y = 1 and a 3x3 matrix B that represents reflection in the line 2x + y = 4.  Calculate AB and BA and interpret each as a translation.  Then find a third line (parallel to the other two) such that a reflection C in this line makes the product CBA equal to a reflection in a line through the origin. 
4. Ryan, problem 23 on page 37.
5. Ryan, problem 26 on page 37.
6. Find the 2x2 matrix A for a reflection in the line x = 2y and the matrix B for a reflection in the line x = -2y.  Calculate the product AB and describe its geometrical effect. Draw a diagram
relating the angle associated with AB to the angle between the two given lines.
7. Find the 3x3 matrix A for a reflection in the line x = 2y - 3 and the 3x3 matrix B for a reflection in the line x = -2y + 1.  Calculate the product AB and describe its geometrical effect.  Draw a diagram showing the two given lines and the point that is held fixed by AB.
8. Let A and B the matrices that represent arbitrary reflections, so that A2 = B2 = identity.
Prove algebraically that AB and BA are inverses of one another.  Draw diagrams to illustrate this for one example where A and B are reflections in parallel lines and one example where A and B are reflections in intersecting lines.
9. Determine the 3x3 matrix A for a counterclockwise rotation though 37 degrees about the point
x = 20, y = 15. You can view this as an eigenvector problem, but perhaps an easier approach is
to determine geometrically the effect of this rotation on the origin.

10. Here is a concrete example of calculating the product of two rotations by the technique shown
on the course Web site under Interactive->Euclidean isometries, the link
"Combine two rotations about different points."
a. Show that reflection in the x-axis followed by reflection in a line of slope 1/3 leads to a counterclockwise rotation through 37 degrees, while reflection in the x-axis followed by reflection in a line of slope 1/2 leads to a counterclockwise rotation through 53 degrees.
b. Determine the geometrical effect of a counterclockwise rotation B through 37 degrees about the point x = 1, y = 3 followed by a counterclockwise rotation A through 53 degrees about the point x = 6, y = 3. Determining the angle for the composition AB is easy. Finding the fixed point is also easy if you use the geometrical approach on the Web site.
11. Determine the geometrical effect of a counterclockwise rotation B through 37 degrees about the point x = 1, y = 0 followed by a clockwise rotation A through 37 degrees about the point x = 5, y = 0.