# Conversion Between Coordinate Systems

1. ### hp-p00nst3r

29
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?

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. ### wbrigg

16
from reading the question it looks like there should be a diagram included...

3. ### hp-p00nst3r

29
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.

4. ### HallsofIvy

40,800
Staff Emeritus
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 $\theta$ degrees is given by
$u= p cos(\theta)+ q sin(\theta)$
$v= -p sin(\theta)+ q cos(\theta)$

5. ### hp-p00nst3r

29
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).

6. ### wbrigg

16
the biggest problem i'm having looking at the possible answers is that none of them contain y, they should be of the form

$$u=xCos\theta +ySin\theta + a$$

sure, the question says with respect to x, but there's a constant at the end, you can't just ignore the y components...

7. ### hp-p00nst3r

29
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.

8. ### hp-p00nst3r

29
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?

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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