Assignment Due Friday, 11/16

Read in Brown: Start reading Chapter 1.  We have moved quickly through sections 1.1, 1.2, 1.3, which should be familiar material for the most part (but you should read them quickly to check).  In class Wednesday, we presented section 1.4 (Isometries) as well as part of 1.6.  We will get to 1.5 soon.  In lab, we will be working on parts of 1.7 and 1.8, so in all you should read Sections 1.4, 1.6-1.8.

Part 1.  Parallel Lines

Suppose that m and n are parallel lines, at distance d apart. M is reflection in line m, and N is reflection in line n

Problem 1.  For any point P, describe as exactly as possible the location of NM(P), describing direction and distance.  Give good reasons.

Problem 2.  Choose coordinate axes so that m and n become vertical lines parallel to the y-axis.  Suppose m is given by x = a, and n is given by x = b.  

• Then if P = (x,y), give a formula for M(P) and N(P).
• For P = (x,y), give a formula for NM(P).
• For P = (x,y), give a formula for MN(P).
• Let u and v be additional lines with U, V the reflections in u and v.  If the line u has equation x = c, what is a line v so that VU = NM?
• Finally, for P as before, compute the formula for UNM(P).

Problem 3.  Continuing with M and N as before, answer these questions as completely as possible (i.e., what kind of isometry, but also which one exactly: For a reflection, what is the mirror? For a translation, what is the direction and magnitude?)

• What isometry is the composition MN?
• What is NM?
• What is (MN)(NM)?
• What is MNM?

Part 2.  Intersecting Lines

Let m and n be lines intersecting at O. Let M be reflection in line m and N be reflection in line n (not to be confused with the points M and N). 

Problem 4.  For any point P, describe as exactly as possible the location of NM(P), describing angle and distance.  Give good reasons.

Problem 5.  Choose polar coordinates (distance r and angle t; look it up if needed) so that m and n become lines through the origin.  Suppose m is given by t = a, and n is given by t = b.  

• Then if P is given by r = h, t = s, give a formula for the polar coordinates of M(P) and N(P). (Hint:  Look for analogies with Problem 2.
• For P above, give a formula for NM(P).
• For P above, give a formula for MN(P).
• Let u and v be additional lines with U, V the reflections in u and v.  If the line u has equation t = c, what is a line v so that VU = NM?
• Finally, for P as before, compute the formula for UNM(P).

Problem 6.  Continuing with M and N as before, answer these questions as completely as possible (i.e., what kind of isometry, but also which one exactly: For a reflection, what is the mirror? For a translation, what is the direction and magnitude?)

• What isometry is the composition MN?
• What is NM?
• What is (MN)(NM)?
• What is MNM?