 #1
Redwaves
 134
 6
 Homework Statement:
 Finding the transformation of a matrix
 Relevant Equations:

##\begin{bmatrix}
cos \theta & sin \theta \\
sin \theta & cos \theta
\end{bmatrix}##
I have the matrix above and I have to find which transformation is that.
##\begin{bmatrix}
cos \theta & sin \theta \\
sin \theta & cos \theta
\end{bmatrix}##
For a vector ##\vec{v}##
##v_x' = v_x cos \theta + v_y sin \theta##
##v_y' = v_x sin \theta  v_y cos \theta##
If ##\phi## is the angle between xaxis and the vector ##\vec{v}##, then ##v_x = r cos \theta ## and ##v_y = r sin \theta##
Thus,
##v_x = r cos \phi cos \theta + r sin\phi sin\theta## = ##r cos(\phi  \theta)##
##v_y = r cos \phi sin \theta  r sin\phi cos\theta## =  ##r sin(\phi  \theta)##
From that, the transformation seems to be an rotation of ##\phi  \theta## clockwise and then a reflection over the x axis. Is this correct?
##\begin{bmatrix}
cos \theta & sin \theta \\
sin \theta & cos \theta
\end{bmatrix}##
For a vector ##\vec{v}##
##v_x' = v_x cos \theta + v_y sin \theta##
##v_y' = v_x sin \theta  v_y cos \theta##
If ##\phi## is the angle between xaxis and the vector ##\vec{v}##, then ##v_x = r cos \theta ## and ##v_y = r sin \theta##
Thus,
##v_x = r cos \phi cos \theta + r sin\phi sin\theta## = ##r cos(\phi  \theta)##
##v_y = r cos \phi sin \theta  r sin\phi cos\theta## =  ##r sin(\phi  \theta)##
From that, the transformation seems to be an rotation of ##\phi  \theta## clockwise and then a reflection over the x axis. Is this correct?
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