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Find the 3 ×3 matrix that produces the described composite 2D transformation​ below, using homogeneous coordinates. Translate by ​( −6​, 3​), and then scale the​ x-coordinate by 0.3 and the​ y-coordinate by 1.3.

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sadnuts112

Answer:  

               「0.8   0   -2

                    0    1.2   3

                    0     0     1 」

Step-by-step explanation:

Cricetus

The [tex]3\times 3[/tex] matrix will be "[tex]\left[\begin{array}{ccc}0.3&0&-1.8\\0&1.3&3.9\\0&0&1\end{array}\right][/tex]".

According to the question,

The [tex]3\times 3[/tex] with homogenous coordinates is:

[tex]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

Translation from (0, 0) to (a, b) will be:

[tex]T = \left[\begin{array}{ccc}1&0&a\\0&1&b\\0&0&1\end{array}\right][/tex]

Here,

  • a = -6
  • b = 4

Translation by (-5,4)

[tex]T = \left[\begin{array}{ccc}1&0&-6\\0&1&3\\0&0&1\end{array}\right][/tex]

Now,

  • Let's find the scaling matrix 5.
  • Scale the x-coordinated by "0.3" and y-coordinates by "1.3".

i.e.,

[tex]S = \left[\begin{array}{ccc}0.3&0&0\\0&1.3&0\\0&0&1\end{array}\right][/tex]

then,

The [tex]3\times 3[/tex] matrix will be:

→ [tex]A = ST[/tex]

      [tex]= \left[\begin{array}{ccc}0.3&0&0\\0&1.3&0\\0&0&1\end{array}\right] \ \left[\begin{array}{ccc}1&0&-6\\0&1&3\\0&0&1\end{array}\right][/tex]

      [tex]= \left[\begin{array}{ccc}0.3&0&-0.3\times 6\\0&1.3&1.3\times 3\\0&0&1\times 1\end{array}\right][/tex]

      [tex]= \left[\begin{array}{ccc}0.3&0&-1.8\\0&1.3&3.9\\0&0&1\end{array}\right][/tex]

Thus the above answer is correct.

       

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