Question

Show that a 2d reflection through x-axis followed by a 2d reflection through the line y=-x is equivalent to pure rotation about origin

Answer 1

Answer: Hence proved

Explanation:

Answer 2

Answer:

Hence prove

E xplanation:

Reflection axis as diagonal y=-x accomplished with $x^{\prime}=-y$

$y^{\prime}=-x$

i). Clockwise Rotation through $45^{\circ}$

ii) ReFlect about y-axis

iii) Counter (lockwise through $45^{\circ}$

Here, The Resultant Transformation matrix is

$\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$

Then, we have prove that 2 D Reflection through The z-axis, followed by a 2 D Reflection through the line y=-x

Equivalent to pure rotation about  

the origin.

$\rightarrow$ Transformation that distorts the shape of an object Such that the transformed shape appears as if the object were composed of internal layers that had been caused to Slide over Each other is called a shear.

$\rightarrow$Two-dimensional transformation Can be Represented in a uniform way by $3 \times 3 matrix$

The $3 * 3$ Matrix that Represent the translation transformation is:

$\left[\begin{array}{lll}x^{\prime} & y^{\prime} & 1\end{array}\right]$=$\left[\begin{array}{lll}x & y & 1\end{array}\right]$$\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\d x & d y & 1\end{array}\right]$

$\text { The } 3 * 3 \text { matrix that Represent the rotation }$

transformation matrix is: -

$\left[\begin{array}{lll}x^{\prime} & y^{\prime} & 1\end{array}\right]$=$\left[\begin{array}{lll}x & y & 1\end{array}\right]$$\left[\begin{array}{ccc}\cos (\theta) & -\sin (\theta) & 0 \\\sin (\theta) & \cos (\theta) & 0 \\0 & 0 & 1\end{array}\right]$

The $3 \times 3$ matrix that Represents the Reflection about the y-axis flips x co-ordinates.

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

$\rightarrow$The 2D Reflection through the z axis, followed by 2D Reflection through the line y=-x by rotating it about the origin

$\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$

a 2D Reflection through the z-axis, followed by a 2 D Reflection through the line y=-x is Equivalent a pure rotation about the origin

#SPJ2