- #1

Redwaves

- 134

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- 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 x-axis 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 x-axis 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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