A quick math question

[NaOH]

"A vector x in R^2 is rotated twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify these two expressions are equivalent."

All I could come up with is R ^ 2 * x = y and R * (R * x) = y

 

[cheesygoldfish]

Originally posted by: NaOH
"A vector x in R^2 is rotated twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify these two expressions are equivalent."

All I could come up with is R ^ 2 * x = y and R * (R * x) = y

damn i wish i was half as smart as you

 

[NaOH]

hmm...i sense sarcasm =(

 

[James Bond]

Originally posted by: NaOH
hmm...i sense sarcasm =(

I think he was serious.

 

[2Xtreme21]

Cheeseygoldfish's posts have been nothing but troll posts. Look at any other thread in which he posts-- just ignore him.

With that said, I unfortunately can't help you. Don't they have websites that will allow you to simulate these types of situations?

 

[NaOH]

Originally posted by: 2Xtreme21
Cheeseygoldfish's posts have been nothing but troll posts. Look at any other thread in which he posts-- just ignore him.

With that said, I unfortunately can't help you. Don't they have websites that will allow you to simulate these types of situations?

I've searched around on the topic and can't find any that explain it thoroughly enough (or simple enough) for me to get a grasp on the concept.

 

[Fenixgoon]

Originally posted by: NaOH
"A vector x in R^2 is rotated twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify these two expressions are equivalent."

All I could come up with is R ^ 2 * x = y and R * (R * x) = y

examine what happens to your basis vectors, [1,0] and [0,1] when you rotate them by the angle theta. this rotation can be expressed as [T]^2 (T being the transformation for rotation by angle theta) or simply by carrying through with T^2 (multiply T by itself and then use that matrix as the second answer). or at least that's how i'd do it.

 

[NaOH]

[T] = [cos(theta) -sin(theta); sin(theta) cos(theta) ] right?

 

[Fenixgoon]

Originally posted by: NaOH
[T] = [cos(theta) -sin(theta); sin(theta) cos(theta) ] right?

if [1;0] is y= 1 and [0,1] is x=1, then yeah, looks good

 

[NaOH]

so one answer would be T^2 * x and the other would be just T^2? I'm not sure if I follow.

 

[randomint]

here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

 

[NaOH]

Originally posted by: randomint
here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

that makes sense to me. Did u write this in paint with ur mouse?

 

[randomint]

Originally posted by: NaOH

Originally posted by: randomint
here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

that makes sense to me. Did u write this in paint with ur mouse?

sadly yes

 

[NaOH]

thanks for taking the time to make an attempt at explaining this to me.

 

[oboeguy]

Search for Givens Rotations, problem solved.

 

[randomint]

Originally posted by: NaOH


thanks for taking the time to make an attempt at explaining this to me.

anytime ... er.. scratch that

 

[NaOH]

Originally posted by: randomint

Originally posted by: NaOH

Originally posted by: randomint
here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

that makes sense to me. Did u write this in paint with ur mouse?

sadly yes

so the two forms should be [x2,y2] = [cos2theta -sin2theta, sin2theta cos2theta] [x,y] and [x2,y2] = [costheta -sintheta, sintheta costheta]^2 [x,y] ?

 

[oboeguy]

The first matrix rotates by 2*theta, while the product rotates by theta twice so that sounds good. You should do the matrix mult yourself as an exercise to show they're the same (use double-angle formulae).

 

[randomint]

Originally posted by: NaOH

Originally posted by: randomint

Originally posted by: NaOH

Originally posted by: randomint
here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

that makes sense to me. Did u write this in paint with ur mouse?

sadly yes

so the two forms should be [x2,y2] = [cos2theta -sin2theta, sin2theta cos2theta] [x,y] and [x2,y2] = [costheta -sintheta, sintheta costheta]^2 [x,y] ?

well you're on the right track as successive rotations mean multiplication of the rotation matrices but multiply them out like oboeguy said to be sure.

 

[potoba]

Originally posted by: NaOH
"A vector x in R^2 is rotated twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify these two expressions are equivalent."

All I could come up with is R ^ 2 * x = y and R * (R * x) = y

which axis are you rotating around? This is simply an example of a transformation/operator you will encounter later on when you hit group theory.

 

[DVK916]

I post math questions, and get almost no help at all, the help I do get is very basic.

 

[TheoPetro]

Originally posted by: DVK916
I post math questions, and get almost no help at all, the help I do get is very basic.

thats because when you ask them you do it in a really, REALLY, annoying way

 

[Kirby]

Originally posted by: TheoPetro

Originally posted by: DVK916
I post math questions, and get almost no help at all, the help I do get is very basic.

thats because when you ask them you do it in a really, REALLY, annoying way

fixed

 

[eLiu]

Originally posted by: NaOH

Originally posted by: randomint

Originally posted by: NaOH

Originally posted by: randomint
here is one expression for the rotation matrix.

c being the final rotated vector and phi being the initial angle the vector was at before it was rotated.

that makes sense to me. Did u write this in paint with ur mouse?

sadly yes

so the two forms should be [x2,y2] = [cos2theta -sin2theta, sin2theta cos2theta] [x,y] and [x2,y2] = [costheta -sintheta, sintheta costheta]^2 [x,y] ?

exactly.

 

[potoba]

lol, another n subsequent polarizer problem