Question

is there anyone that is good with matrices because i need help with this one

Answer 1

Find the image matrix for the following counterclockwise rotation.

Answer 2

Lets start with the transformation matrix for an arbitrary rotation:
\[R =\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]\]

Answer 3

A counterclockwise rotation of 180 degrees will simply be
\[\theta = \pi\]

Answer 4

ok so whats next

Answer 5

\[R =\left[\begin{array}{cc}
\cos \pi & -\sin \pi \\
\sin \pi & \cos \pi
\end{array}\right] =\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right] \]

Answer 6

Now apply the transformation to the coordinate vectors

Answer 7

\[
\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]
\left[\begin{array}{cc}
3 \\
1
\end{array}\right] =
\left[\begin{array}{cc}
-3 \\
-1
\end{array}\right]
\]\[
\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]
\left[\begin{array}{cc}
6 \\
2
\end{array}\right] =
\left[\begin{array}{cc}
-6 \\
-2
\end{array}\right]
\]\[
\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]
\left[\begin{array}{cc}
4 \\
5
\end{array}\right] =
\left[\begin{array}{cc}
-4 \\
-5
\end{array}\right]
\]
So we have the "image matrix"
\[
\left[\begin{array}{cc}
-3 & -6 & -4 \\
-1 & -2 & -5
\end{array}\right]
\]

Answer 8

ooooo ok i seee thank you so much