Are these compositions of linear transformations reflections or rotations?

[mneox]

Homework Statement

if Sa: R2 -> R2 is a rotation by angle a counter-clockwise
if Tb: R2 -> R2 is a reflection in the line that has angle b with + x-axis

Are the below compositions rotations or reflections and what is the angle?
a) Sa ○ Tb
b) Ta ○ Tb

Homework Equations

I don't think you need any for this question.

The Attempt at a Solution

Can somebody explain what this question is asking..? I think I understand the concept of rotations and reflections. But how are you supposed to tell if Sa ○ Tb is a rot or ref?

I know that Sa ○ Tb means you have to do the reflection first and then the rotation, but how do you know if the composition is a rotation or reflection? I'm just confused about that.

And how would you find the angle?

 

[LCKurtz]

Homework Statement

if Sa: R2 -> R2 is a rotation by angle a counter-clockwise
if Tb: R2 -> R2 is a reflection in the line that has angle b with + x-axis

Are the below compositions rotations or reflections and what is the angle?
a) Sa ○ Tb
b) Ta ○ Tb

Homework Equations

I don't think you need any for this question.

The Attempt at a Solution

Can somebody explain what this question is asking..? I think I understand the concept of rotations and reflections. But how are you supposed to tell if Sa ○ Tb is a rot or ref?

I know that Sa ○ Tb means you have to do the reflection first and then the rotation, but how do you know if the composition is a rotation or reflection? I'm just confused about that.

And how would you find the angle?

I think you do need the equations for reflection and rotation. Apply them both in the orders given and look at them. The form of the equations will tell you if they are a rotation or reflection.

 

[mneox]

I think you do need the equations for reflection and rotation. Apply them both in the orders given and look at them. The form of the equations will tell you if they are a rotation or reflection.

How do I apply them w/o vectors??

 

[LCKurtz]

How do I apply them w/o vectors??

Hard to answer without seeing what you have. What do you have for the equations of reflection or rotation? What have you tried?

 

[mneox]

Hard to answer without seeing what you have. What do you have for the equations of reflection or rotation? What have you tried?

I know rotation goes like

Rot$$\theta$$ = [cos$$\theta$$ -sin$$\theta$$; sin$$\theta$$ cos$$\theta$$]

then reflection goes like

Ref$$\theta$$ = [cos2$$\theta$$ sin2$$\theta$$; sin2$$\theta$$ -cos2$$\theta$$]

So since it was a composition Sa ○ Tb for the first one, i tried to multiply the matrices for rot and ref together.. but that only gave me some crazy looking answer that I don't know how to interpret. :(

 

[LCKurtz]

I know rotation goes like

Rot$$\theta$$ = [cos$$\theta$$ -sin$$\theta$$; sin$$\theta$$ cos$$\theta$$]

then reflection goes like

Ref$$\theta$$ = [cos2$$\theta$$ sin2$$\theta$$; sin2$$\theta$$ -cos2$$\theta$$]

So since it was a composition Sa ○ Tb for the first one, i tried to multiply the matrices for rot and ref together.. but that only gave me some crazy looking answer that I don't know how to interpret. :(

Those look good. So use them with appropriate a or b. The "crazy looking" answers might look better if you review the trig addition formulas for sines and cosines. You know, sin(a+b) = ? etc.

 

[HallsofIvy]

For very general questions like this, it might help to look a special examples. Suppose the angle of rotation is 90 degrees and the line of reflection is the x- axis. Then what does S○ T do to (1, 0) and (0, 1)? T(1, 0) is (1, 0) and S(1, 0) is (0, 1). T(0, 1)= (0, -1) and S(0, -1)= (1, 0). That is, S○ T(1, 0)= (0, 1) and S○ T(0, 1)= (1, 0). That looks like a reflection (in the line y= x) to me.

Take T1 to be reflection in the x- axis and T2 to be reflection in the y-axis. Then T1(1, 0)= (1, 0) and T2(1, 0)= (-1, 0). T1(0, 1)= (0, -1) and T2(0, -1)= (0, -1). That is, T2○ T1(1, 0)= (-1, 0) and T2○ T1(0, 1)= (0, -1). That looks like a rotation (through 180 degrees) to me.