Homework for weeks 1 and 2

Homework will always be collected on Thurdays and (usually) returned on Tuesdays. The problems listed under ``Do'' are to help you do the material. Don't hand them in. Do hand in the ones under ``Hand in''.

``Extensions'' are beyond-the-basics problems. To get an A in the course, you must do some extensions. Extensions may also include good proofs or generalizations of assigned problems.

Due date	Do	Hand in
Thurs. April 8	(Read Ch. 1)	Problem A below
Thurs. April 15	Ch. 1, problems 1-6, 9-11, 13	Problems B and C below

A.

This activity was done in class: Construct two lines intersecting at $45^\circ$. Draw a capital letter R anywhere on the paper. Reflect the R across the first line; then reflect all the Rs across the second line. Then reflect all the Rs across the first line again. Continue until you don't get anythng new. How many Rs are there in all? How are they related to each other?

(a)

If you did the Rs activity with a $150^\circ$ angle, how many Rs would you get?

(b)

If you did the Rs activity with a $142.5^\circ$ angle, how many Rs would you get?

(c)

If you did the Rs activity with a $\pi^\circ$ angle (not $\pi$ radians), how many Rs would you get?

(d)

(Extension) Generalize to any angle.

B.

Let f((x,y)) = (-y+5,x+1) and consider the lines L_i defined by equations

$$L_1: \ \ Y=2;\ \ \ \ L_2: \ \ Y=3;\ \ \ \ L_3: \ \ X=1;\ \ \ \ L_4: \ \ Y=X;\ \ \ \ L_5: \ \ Y=X+1$$

(a)

Prove that f is a collineation.

(b)

Find the equations of the images of the lines L_i under f.

(c)

Graph and label the lines L_i and their images on graph paper.

(d)

Describe what this mapping does to the plane geometrically and (extension) prove it.

C.

Prove that lines with equations aX+bY+c=0 and dX+eY+f=0 are parallel if and only if ae-bd=0. Use the definition given in the book: two coplanar lines are parallel if they are identical or if they do not intersect.