Linear Transformation: find dilating/rotation matrix

[bcahmel]

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.

 

[Ray Vickson]

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
$$\pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.$$
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
$$\pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.$$
Note that the two transformation matrices are transposes of each other.

RGV

 

[vela]

The way you combined the rotation and dilation isn't correct. As Ray said, do one first and then the other.

 

[bcahmel]

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.

 

[LCKurtz]

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.

 

[bcahmel]

Thank you LCKurts, vela, and Ray! So to summarize: think of the transformations separately, and then multiply the two transformation matrices together.
Thanks again