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Which homogeneous $2D$ matrix transforms the figure (a) on the left side to figure (b) on the right?

  1. $\begin{pmatrix} 0 & 2 & -6 \\ 2 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$
  2. $\begin{pmatrix} 0 & -2 & 6 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$
  3. $\begin{pmatrix} 1 & -2 & 6 \\ 1 & 0 & 2 \\ 0 & 0 & 1 \end{pmatrix}$
  4. $\begin{pmatrix} 0 & 2 & 6 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$
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Answer is option (B)

Figure on the right is rotated anti-clock wise by 90 degree.

As we know the Rotation matrix in case of anti clock wise rotation is

Option are given in 3D matrix, so we have to transform given 2D matrix into 3D homogenous matrix system.

So it is like,                    $\begin{pmatrix} cos90^{\circ} & -sin90^{\circ}& 0 \\ sin90^{\circ}&cos90^{\circ} & 0\\ 0& 0& 1 \end{pmatrix} = \begin{pmatrix} 0 &-1& 0 \\ 1&0 & 0\\ 0& 0& 1 \end{pmatrix}$

Here, we can see in figure rotation is done with respect to 2, so options are already given after rotating it w.r.t 2. We don't have need to do it again. So, matrix after rotation w.r.t 2 looks like \begin{pmatrix} 1 &-2& 0\\ 1&0& 0\\ 0& 0& 1 \end{pmatrix}

Upto this by seeing position of - sign we can easily find that correct option is either B or C. So, lets find it among these two now.

We know that Translation matrix is given by  \begin{pmatrix} 1 &0& tx \\ 0&1 & ty\\ 0& 0& 1 \end{pmatrix}

See the figure the point which is at (0,0) is now transformed to 6 along x-axis and 1 along y-axis

Therefore, tx = 6 & ty = 1, putting values we can get the matrix

                                                          $\begin{pmatrix} 1 & -2& tx \\ 1&0& ty\\ 0& 0& 1 \end{pmatrix} = \begin{pmatrix} 0 &-2& 6 \\ 1&0 & 1\\ 0& 0& 1 \end{pmatrix}$

Hence, Option (B) is correct.

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