Rotation of cartesian coordinate system

In summary: It should be ycos(theta) + xsin(theta). This is because the y-coordinate is being rotated by theta, but it is also multiplied by the cosine of beta, which is the x-coordinate in the original system. So, the correct formula for y' is ycos(theta) + xsin(theta).
  • #1
xzibition8612
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Homework Statement


Please see the rotation formula in the attachment.

Homework Equations





The Attempt at a Solution


I understand this formula rotates x,y into x',y' by some angle theta. Problem is, how is this formula derived? I cannot for the life of me visualize the cosine and sine transformation physically. Can someone explain to me how you get this formula. Thank you very much.
 

Attachments

  • cartesian counterclockwise rotation.doc
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  • #2
Consider the vector that extends from the origin to the point (x,y) in the base coordinate system. It has some magnitude R and angle β with respect to the x-axis of the coordinate system. In fact, x = Rcos(β) and y = Rsin(β).

Rotating that point around the origin by some angle θ is equivalent to rotating the vector by θ, so what would the coordinates of its endpoint be?
 
  • #3
so the end points would be x=Rcos(β+θ), y=Rsin(β+θ). Then what? I still can't see how this relates to the formula, espcially how in the formula for x' and y' individually there are x and y terms together.
 
  • #4
What you just found are the [itex]x'[/itex] and [itex]y'[/itex] coordinates. Expand the sines and cosines using angle sum formulas and put any sines or cosines of [itex]\beta[/itex] in terms of the origina [itex]x,y[/itex].
 
  • #5
x' = R(cosβcosθ-sinβsinθ)
y' = R(sinβcosθ+sinβcosθ)

x' = R[(x/R)cosθ-(y/R)sinθ]
y' = R[(y/R)cosθ+(y/R)cosθ]

arrrrgh almost there. First term in y' is wrong. I get y' = ycosθ ... instead of y' = xsinθ ...Can someone point out myt mistake? Thanks a lot for your help!
 
  • #6
In your second line, you forgot to switch beta and theta in the second term.
 

1. What is a cartesian coordinate system?

A cartesian coordinate system is a mathematical tool used to locate points in a two-dimensional plane. It consists of two perpendicular lines, the x-axis and y-axis, which intersect at a point called the origin.

2. How does rotation affect the cartesian coordinate system?

Rotation of the cartesian coordinate system involves rotating the axes around the origin by a certain angle. This changes the position of points on the plane and alters the values of their coordinates.

3. What are the formulas for rotating a cartesian coordinate system?

The formulas for rotating a cartesian coordinate system are:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
Where x' and y' are the new coordinates after rotation, x and y are the original coordinates, and theta is the angle of rotation.

4. How is rotation of the cartesian coordinate system used in real life?

Rotation of the cartesian coordinate system is used in many real-life applications, such as computer graphics, robotics, and navigation systems. For example, in computer graphics, rotation is used to create 3D images by changing the position of points on a 2D plane. In robotics, it helps in controlling the movement of robotic arms. In navigation systems, it is used to determine the direction and location of objects.

5. What are the advantages of using a cartesian coordinate system for rotation?

One of the main advantages of using a cartesian coordinate system for rotation is that it allows for precise and accurate calculations. It also provides a visual representation of the rotated points, making it easier to understand and interpret the results. Additionally, the formulas for rotation are simple and can be easily applied in various fields of science and engineering.

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