Linear Transformation: find dilating/rotation matrix

  • Thread starter bcahmel
  • Start date
  • #1
bcahmel
25
0

Homework Statement



The vector A has length 8.5, and makes an angle of 5pi/19 with the x-axis.

The vector B has length 6, and makes an angle of 8pi/19 with the x-axis.

Find the matrix which rotates and dilates vector into vector .

Homework Equations


Rotation matrix in counterclockwise direction: \begin{equation}

\left[
\begin{array}{ccc}
cos∅ & -sin∅\\
sin∅ & cos∅\\
\end{array}
\right]

\end{equation}

Dilation matrix:
\begin{equation}

\left[
\begin{array}{ccc}
c & 0\\
0 & c\\
\end{array}
\right]

\end{equation}


The Attempt at a Solution



\begin{equation}

\left[
\begin{array}{ccc}
(12/17)cos(3pi/19) & -sin(3pi/19)\\
sin(3pi/19) & (12/17)cos(3pi/19)\\

\end{array}
\right]

\end{equation}

My line of thinking: to move A into B, it needs to move counterclockwise from its current positions by an angle of 3pi/19. A needs to "shrink" by a factor of 12/17.

However, this is incorrect. I would really appreciate it if someone could point me in the right direction, or even give me a similar example. Thank you.
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement



The vector A has length 8.5, and makes an angle of 5pi/19 with the x-axis.

The vector B has length 6, and makes an angle of 8pi/19 with the x-axis.

Find the matrix which rotates and dilates vector into vector .

Homework Equations


Rotation matrix in counterclockwise direction: \begin{equation}

\left[
\begin{array}{ccc}
cos∅ & -sin∅\\
sin∅ & cos∅\\
\end{array}
\right]

\end{equation}

Dilation matrix:
\begin{equation}

\left[
\begin{array}{ccc}
c & 0\\
0 & c\\
\end{array}
\right]

\end{equation}


The Attempt at a Solution



\begin{equation}

\left[
\begin{array}{ccc}
(12/17)cos(3pi/19) & -sin(3pi/19)\\
sin(3pi/19) & (12/17)cos(3pi/19)\\

\end{array}
\right]

\end{equation}

My line of thinking: to move A into B, it needs to move counterclockwise from its current positions by an angle of 3pi/19. A needs to "shrink" by a factor of 12/17.

However, this is incorrect. I would really appreciate it if someone could point me in the right direction, or even give me a similar example. Thank you.

You can either rotate first, then shrink, or shrink first, then rotate. Also, be careful about rotations: for rotation a coordinate system through angle θ but leaving a vector v = <x,y> unchanged, the coordinates x' and y' of v in the new coordinate system are given by
[tex] \pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & -\sin(\theta)\\
\sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.[/tex]
However, if we rotate the vector v through angle θ but keeping the coordinate system unchanged, the coordinates x' and y' of the new vector v' are given by
[tex] \pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & \sin(\theta)\\
-\sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.[/tex]
Note that the two transformation matrices are transposes of each other.

RGV
 
  • #3
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,761
2,400
The way you combined the rotation and dilation isn't correct. As Ray said, do one first and then the other.
 
  • #4
bcahmel
25
0
To dilate:\begin{equation}

\left[
\begin{array}{ccc}
(12/17) & 0\\
0 & (12/17) \\

\end{array}
\right]

\end{equation}

To rotate:
\begin{equation}

\left[
\begin{array}{ccc}
cos(3pi/19) & -sin(3pi/19)\\
sin(3pi/19) & cos(3pi/19)\\

\end{array}
\right]

\end{equation}


Does the above look right?

So I'm trying to find matrix, a 2X2 matrix so that: \begin{equation}

\left[
\begin{array}{ccc}
A\\
\\

\end{array}
\right]

*

\left[
\begin{array}{ccc}
8.5cos(5pi/19)\\
8.5sin(5pi/19)\\

\end{array}
\right]

=

\left[
\begin{array}{ccc}
6cos(8pi/19)\\
6sin(8pi/19)\\

\end{array}
\right]

\end{equation}

Is there a way to put these two transformations together, into one matrix, A? Thank you for the help you have given so far.
 
  • #5
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,568
774
So I'm trying to find matrix, a 2X2 matrix so that: \begin{equation}

\left[
\begin{array}{ccc}
A\\
\\

\end{array}
\right]

*

\left[
\begin{array}{ccc}
8.5cos(5pi/19)\\
8.5sin(5pi/19)\\

\end{array}
\right]

=

\left[
\begin{array}{ccc}
6cos(8pi/19)\\
6sin(8pi/19)\\

\end{array}
\right]

\end{equation}

In your original post, if you multiply your dilation matrix by your rotation matrix correctly, you should have a 12/17 times all four entries. Then try that for your A.
 
  • #6
bcahmel
25
0
Thank you LCKurts, vela, and Ray! So to summarize: think of the transformations separately, and then multiply the two transformation matrices together.
Thanks again
 

Suggested for: Linear Transformation: find dilating/rotation matrix

  • Last Post
Replies
1
Views
319
  • Last Post
Replies
4
Views
413
Replies
16
Views
812
Replies
9
Views
364
Replies
10
Views
612
Replies
3
Views
600
Replies
1
Views
691
  • Last Post
Replies
2
Views
163
Replies
5
Views
1K
Top