Conversion Between Coordinate Systems

  1. Hello, I've been stuck on this problem for awhile and I've tried googling up some solutions but I still cannot find an answer to this question.

    1. The problem statement, all variables and given/known data
    An x-y coordinate system is shown below. A second system, u-v, is also shown. What is the relationship between the u-coordinate and the x-coordinate?
    [​IMG]
    A. u = -0.57x + 3.44
    B. u = 0.57x - 3.44
    C. u = -1.74x + 10.44
    D. u = 1.74x - 10.44
    E. None of the above

    2. Relevant equations
    I am not quite sure which equations are relevant but I think it has to do with linear transformations because of the rotating axes.


    3. The attempt at a solution
    I have tried applying the rotation matrix with a theta of 125 deg which gives me
    [cos(125deg) -sin(125deg)
    sin(125deg) cos(125deg)]
    and I find that the x coordinate of the resulting transformation is T(x) = x*cos(125) - y*sin(125). Since in the question, it asks for the x coordinate only, I would assume that y = 0. After, I subbed (x-6) into x to translate the coordinate to match the picture. So my final answer would be: -x*cos(125) + 3.44. It's one of the choices, but it's not the right answer.

    Thank you. Any help is appreciated.
     
  2. jcsd
  3. from reading the question it looks like there should be a diagram included...
     
  4. Ah, my mistake. I did post an image but I guess I was the only one who could see the link since I was logged in to the school's online course website. I thought everyone else could access the picture. I've uploaded the picture on imageshack now and edited the first post.
    Thanks for pointing that out.

    EDIT: for some reason I can't edit my first post. So here's the image.
    [​IMG]
     
  5. HallsofIvy

    HallsofIvy 40,308
    Staff Emeritus
    Science Advisor

    First, of course, you have to translate the origin. Let "p, q" represent a coordinate system with axes parallel to the x,y axes but with origin at x= 6, y= 3. Then p= x-6, q= y- 3.

    Now, the u, v coordinate system is just the p,q system rotated by 35 degrees. In general, the p,q system rotated by [itex]\theta[/itex] degrees is given by
    [itex]u= p cos(\theta)+ q sin(\theta)[/itex]
    [itex]v= -p sin(\theta)+ q cos(\theta)[/itex]
     
  6. Isn't it a 125 degree rotation since the x axis is pointing downwards? It has to rotate a total of 125 degrees.

    According to the answer key, the answer is C. u = -1.74x + 10.44.

    I subbed p = x - 6 and q = y - 3 into the u = equation, but im not getting -1.74x at all.
    I got u = (x - 6) cos (35) + (y - 3)sin(35).
     
  7. the biggest problem i'm having looking at the possible answers is that none of them contain y, they should be of the form

    [tex]u=xCos\theta +ySin\theta + a[/tex]

    sure, the question says with respect to x, but there's a constant at the end, you can't just ignore the y components...
     
  8. I found that strange as well
    If i try just regular trial and error on the answers, it still doesn't make much sense to me. If i sub x = 6 into choice C, it gives u = 0. That doesn't really make sense on the diagram.
     
  9. I think I answered the question.
    Here is my work. Can you guys please take a look and tell me if my work is right?
    [​IMG]
    Instead of working with xy and uv, I limited it only to the x and u components
    From what the question is like, I think we are not working with the entire coordinate system at all. It is as if the y and v part of the coordinate systems do not exist at all.
     
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