September 13th, 2018, 07:33 PM  #1 
Newbie Joined: Aug 2018 From: United States Posts: 2 Thanks: 0  geometric transformations
Matt wants to reflect a shape over the xaxis and then reflect it over the yaxis. Cathy says that this would be the same as rotating the shape 180 degrees. Do you agree or disagree with Cathy? Justify your answer and use matrix transformations as part of your justification. thanks for any help!! 
September 13th, 2018, 07:54 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,262 Thanks: 1198 
suppose we have a point $(x,y)$ Reflecting it would be accomplished by premultiplying by the matrix $ref_x = \begin{pmatrix}1 &0 \\ 0 &1\end{pmatrix}$ similarly $ref_y = \begin{pmatrix}1 &0 \\ 0 &1\end{pmatrix}$ multiplying these two reflections matrices $ref_y ref_x = \begin{pmatrix}1 &0 \\ 0 &1 \end{pmatrix}$ a rotation matrix of $\pi$ radians is given by $rot_\pi = \begin{pmatrix}\cos(\pi) &\sin(\pi) \\ \sin(\pi) &\cos(\pi) \end{pmatrix} =\begin{pmatrix}1 &0 \\ 0 &1 \end{pmatrix}$ and we see that the two reflections are indeed equivalent to the single rotation by $\pi$ 
September 13th, 2018, 09:17 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,089 Thanks: 1902 
Your $ref_x$, $ref_y$ and $rot$ are usually denoted by $\operatorname{Ref_y}$, $\operatorname{Ref_x}$ and $\operatorname{Rot}$ respectively. If the point $(x,~ y)$ is written as a row vector, use postmultiplication. If the point $(x,~ y)$ is written as a column vector, use premultiplication. 

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