Solving Seitz Symbol & E(2) Homework

[83956]

Homework Statement

Let (A l b) be an element of E(2)

(a) Show that (A l b) is a rotation if A is a nontrivial rotation (show that it fixes some point).
(b) Show that (A l b) is a glide reflection if A is a reflection (find the line, not necessarily through the origin, that is taken into itself - draw a picture).
(c) Conclude that every element of E(2) is a rotation, a glide reflection, or a translation.

Homework Equations

The Attempt at a Solution

(a) A nontrivial rotation takes a vector that starts at the origin to another vector that starts at the origin. So (0,0) is the fixed point?
The matrix first line: cos(theta) -sin(theta) second line: sin(theta) cos(theta)]
is a rotation. So to show that (A l b) is a rotation , is A the matrix above, x a point in the plane E(2) and b the vector first line: 1 second line: 0??

I need to see this worked out so the light bulb will click!

 

[lippyka]

(b) A reflection in E(2) is a matrix first line: -1 0 second line: 0 1. So for (A l b) to be a glide reflection the matrix A has to be the one above. But I don't understand how to find the line that is taken into itself. (c) Can someone explain this part?