How Do Rotational Transformations Affect Coordinates?

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Homework Help Overview

The discussion revolves around understanding how rotational transformations affect coordinates, specifically in the context of a geometric representation involving rulers and their lengths. The original poster seeks clarification on the transition from the invariance of length under rotation to the equations representing the rotated coordinates.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the invariance of length under rotation and question how to derive the transformation equations for coordinates. There is an exploration of geometric interpretations, including the use of trigonometric functions and the relationship between the components of the coordinates.

Discussion Status

The discussion is active, with participants sharing insights and asking questions about the geometric setup and the trigonometric relationships involved. Some guidance has been offered regarding the use of a circle and the roles of the components in the transformation equations.

Contextual Notes

Participants are working within the constraints of a classroom setting, referencing a lecture and the need to understand the derivation of the transformation equations without explicit solutions being provided.

Fusilli_Jerry89
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Just a quick question:

In my prof's lecture he drew two rotated rulers of the same length and showed that length is invariant under rotations. Obviously x^2 + y^2 = x'^2 + y'^2 = length of ruler, but how do we go from this to:

x' = xcos(theta) + ysin(theta)
y' = ycos(theta) - xsin(theta)

That's what my prof wrote down..
 
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Fusilli_Jerry89 said:
In my prof's lecture he drew two rotated rulers of the same length and showed that length is invariant under rotations. Obviously x² + y² = x'² + y'² = length of ruler, but how do we go from this to:

x' = xcosθ + ysinθ
y' = ycosθ - xsinθ

Hi Fusilli_Jerry89! :smile:

(have a theta: θ and a squared: ² :smile:)

Just draw a circle with two rulers in it, one along the x-axis, and the other at an angle θ.

Then use trig. :wink:
 
lol i sorry I know this is easy but for some reason I can't figure out where we get the y from. Is that a component? And is it the hypotenuse of something?
 
Fusilli_Jerry89 said:
lol i sorry I know this is easy but for some reason I can't figure out where we get the y from. Is that a component? And is it the hypotenuse of something?

No, y is the other short side of the triangle.

Yes, y is a component, just like x.

The ruler in the circle at an angle θ has its endpoint at x' = 1, y' = 0,

and simple trig shows that that is the same as x = cosθ, y = sinθ. :smile:
 

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