Mathematics 133       Fall 2001        Homework


        Due Monday October 1
            Chapter 1:  Exercises 2iii, 2iv, 5, 9, 10

        Due Wednesday October 3
            Chapter 1:  Exercises 6, 7, 12i, 12ii, 29

        Due Friday October 5
            Chapter 1:  Exercises 8, 11

        Due Monday October 8
            Chapter 1:  Exercises 12iii, 14(last sentence)
            Find the equation of the line that passes through the point 
                (5,2) and is perpendicular (resp. parallel) to the line
                3x + 2y + 10 = 0 ; sketch all three lines.

        Due Wednesday October 10
            Chapter 1:  Exercises 17, 13, 37

        Due Friday October 12
            Chapter 1:  Exercise 22
            Write an explicit formula for the reflection in the line
                4x + 3y + 2 = 0 .

        Due Monday October 15
            Chapter 1:  Exercise 26ii
            Prove the multiplication table on page 28.

        Due Wednesday October 17
            Chapter 1:  Exercises 20, 26i, 26iii

        Due Monday October 22
            Chapter 1:  Exercises 23, 27
            Chapter 2:  Exercise 33

        Due Wednesday October 24
            Chapter 2:  Exercises 5, 6, 7

        Due Friday October 26
            Chapter 2:  Exercise 24
            For each transformation in Exercise 2.1, sketch the image of the
                triangle  ABC , where  A = (0,0), B = (1,0), C = (0,1) .
            Expand  < P-A , P-B > = 0 .
            Expand and simplify  d(P,M) = d(A,B)/2 , where  M  is the
                midpoint of the segment  AB .

        Due Monday October 29
            Chapter 2:  Exercises 1(fixed points only), 11
            Chapter 3:  Exercise 1

        Due Wednesday October 31
            Sketch the following figures and derive composition tables for
                their symmetry groups: the Chevrolet "bowtie", the Nissan
                "bandaid", the Mercedes "star".

        Due Friday November 2
            Chapter 4:  Exercises 1iii, 1iv, 2
            Show, by example, that cross product is neither commutative
                nor associative.

        Due Monday November 5
            Chapter 4:  Exercises 5, 7
            Show that the points  P , Q , R  given in Exercise 4.6 lie on a
                single line and find its poles; also compute  d(P,Q), d(Q,R),
                d(P,R) .

        Due Wednesday November 7
            Chapter 4:  Exercises 9, 10(omit ii, iii), 17, 18i

        Due Friday November 9
            Chapter 4:  Exercise 23
            Find the orthogonal matrix for reflection in the line with pole
                (2/7, 3/7, 6/7) [with respect to the standard basis].  

        Due Friday November 16
            Chapter 4:  Exercises 20, 22, 24, 27

        Due Monday November 19
            Chapter 5:  Exercise 2 (write explicitly: the 4 given points, the
                poles of the 2 named lines, the final intersection point)

        Due Wednesday November 21
            Write the dual to Theorem 5.5 (page 127); illustrate with a figure.
            Prove that the image of a line under a spherical isometry is a line;
                find a pole of the image line in terms of the given data.
            Chapter 4:  Exercise 27 (assuming that the line has pole (0,0,1),
                write the matrix for each transformation in the group)

        Due Monday November 26
            Extend the lines of Exercise 1.11 (page 35) to lines in the 
                projective plane [according to the standard imbedding (x,y)
                = (x:y:1)], compute the intersection point in the projective
                plane, and interpret the result in the affine plane.
            Let  T  be the projective collineation determined by the matrix
                with columns (or rows) (1,1,1), (1,1,0), (1,0,0); let P 
                = (1:2:3) .  Compute  TP  and the pole of the image under
                T  of the line with pole  P .

        Due Wednesday November 28
            Sketch a nondegenerate figure for Theorem 5.5 (page 127) in which
                A_1, A_2, A_3 are collinear.
            Sketch a figure for Theorem 5.5 (page 127) in which A_1, A_2, B_1
                are collinear.
            Chapter 6:  Exercise 1

        Due Friday November 30
            Let P and Q be distinct points of the projective plane.  Show that
                the locus of  d(X,P) = d(X,Q)  is the union of two lines;
                determine their poles if P = (1:0:0), Q = (3:4:0).
            Chapter 7:  Exercise 2

        Due Monday December 3
            Chapter 7:  Exercises 1i, 5i
            Sketch the graph of  s = cosh t .

        Due Wednesday December 5
            Chapter 7:  Exercise 6i
            Let P = (0,0,1), Q = (2,2,3), R = (2,-2,3); find the unit normals
                (poles) of the 3 lines determined by these points; compute 
                d(P,Q), d(Q,R), d(P,R) .

        Due Friday December 7
            Find the matrix for reflection in the line with pole (2,1,2) 
                [with respect to the standard basis].  
            Find the poles of a line through (0,0,1) that intersects the line
                with pole (2,1,2), the poles of both lines through (0,0,1) 
                that are parallel to the line with pole (2,1,2), and the poles
                of a line through (0,0,1) that is ultraparallel to the line 
                with pole (2,1,2).