Question

If the reflection along the line y=x is equivalent to the reflection along the x axis followed by counter clockwise rotation by theta° then find the value of theta

Answer 1

sorry I don't understand bla

Answer 2

The value of θ is 0°.

Explanation:

To find the value of θ, following steps are to be followed:

Transformation matrix for reflection about the line y=x

T1 = $\left[\begin{array}{ccc}0&1\\1&0\end{array}\right]$

Transformation matrix for reflection relative to x-axis

T2 = $\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right]$

Transformation matrix for counter-clockwise rotation

T3 = $\left[\begin{array}{ccc}cos&sin\\-sin&cos\end{array}\right]$

Now, to prove that

T2 = T2 * T3

i.e., $\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right]$ = $\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right]$ $\left[\begin{array}{ccc}cos&sin\\-sin&cos\end{array}\right]$

Now, taking the value of θ = 0 to 90°, we will get

$\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right] = \left[\begin{array}{ccc}1&0\\0&-1\end{array}\right] \left[\begin{array}{ccc}cos&sin\\-sin&cos\end{array}\right]$ = $\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right]$ $\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$ = $\left[\begin{array}{ccc}1&0\\0&-1\end{array}\right]$

Hence, the value of θ is 0°.