- #1
JC2000
- 186
- 16
Moved from a technical forum, so homework template missing
Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction.
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the coordinate system. And the table below for rotation of the axes where (x,y) are the coordinates before rotation :
\begin{array}{|c|c|c|c|}
\hline
& x & y\\ \hline
x'&cos \theta & sin \theta\\ \hline
y'&-sin \theta &cos \theta &\\ \hline
\\ \hline.
\end{array}
Since Rotation followed by Translation of the axes or vice versa should give the same result, I attempted to solve it in multiple ways but did not get the same result.
x,y are the coordinates before transformation while x',y' are coordinates after transformation. h,k represent the coordinates of the origin after translation.)
Case 1 : Translation followed by Rotation
Using the above formulas I got the following results :
$$x'= (x-h)cos \theta + (y-k)sin\theta$$ and $$y' = -(x-h)sin\theta + (y-k)cos\theta$$ ...(A)
OR
$$x-h = x'cos\theta - y'sin\theta$$ and $$y-k = x'sin\theta + y'cos\theta$$ ...(B)
Case 2 : Rotation followed by Translation :
$$x = (x'+h)cos\theta - (y'+k)sin\theta$$ and $$y = (x'+h)sin\theta + (y'+k)cos\theta$$...(C)
OR
$$x'+ h = xcos\theta + ysin\theta A$$ and $$y'+k = -xsin\theta + ycos\theta$$...(D)
Upon plugging in the values (A) and (B) gave matching answers with ##P"(-6,2) ##whereas using (C) and (D) gave ##P"(-5,5)##.
Thus I had the following questions:
1. Are the formulas (A),(B),(C),(D) correct?
2. If not, then why?
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the coordinate system. And the table below for rotation of the axes where (x,y) are the coordinates before rotation :
\begin{array}{|c|c|c|c|}
\hline
& x & y\\ \hline
x'&cos \theta & sin \theta\\ \hline
y'&-sin \theta &cos \theta &\\ \hline
\\ \hline.
\end{array}
Since Rotation followed by Translation of the axes or vice versa should give the same result, I attempted to solve it in multiple ways but did not get the same result.
x,y are the coordinates before transformation while x',y' are coordinates after transformation. h,k represent the coordinates of the origin after translation.)
Case 1 : Translation followed by Rotation
Using the above formulas I got the following results :
$$x'= (x-h)cos \theta + (y-k)sin\theta$$ and $$y' = -(x-h)sin\theta + (y-k)cos\theta$$ ...(A)
OR
$$x-h = x'cos\theta - y'sin\theta$$ and $$y-k = x'sin\theta + y'cos\theta$$ ...(B)
Case 2 : Rotation followed by Translation :
$$x = (x'+h)cos\theta - (y'+k)sin\theta$$ and $$y = (x'+h)sin\theta + (y'+k)cos\theta$$...(C)
OR
$$x'+ h = xcos\theta + ysin\theta A$$ and $$y'+k = -xsin\theta + ycos\theta$$...(D)
Upon plugging in the values (A) and (B) gave matching answers with ##P"(-6,2) ##whereas using (C) and (D) gave ##P"(-5,5)##.
Thus I had the following questions:
1. Are the formulas (A),(B),(C),(D) correct?
2. If not, then why?