Question

the 3×3 matrix obtained from the translation by (3 1) followed by the rotation 45° about the origin​

Answer 1

Answer:

$\left[\begin{array}{ccc}0&1&5\\0&1&2\\1&1&1\end{array}\right]$

Step-by-step explanation:

Given,

3x3 matrix

To Find,

The rotation about fixed point.

Concept:

Rotation about any arbitrary point or fixed point using composite transformation.

So,

Step 1: Translate fixed point to origin.

$(T_{v} )=\left[\begin{array}{ccc}1&0&-h\\0&1&-k\\0&0&1\end{array}\right]$

Step 2: Perform rotation $(R_{0} )$

$(R_{0} )=\left[\begin{array}{ccc}cosx&-sinx&0\\sinx&cosx&0\\0&0&1\end{array}\right]$

(where, x is 45°.)

Step 3: Translate the point back to original point.

$(T_{-v} )= \left[\begin{array}{ccc}1&0&h\\0&1&k\\0&0&1\end{array}\right]$

Now,

The transformation matrix = $[T_{-v}].[R_{0}].[T_{v}]$

= $\left[\begin{array}{ccc}1&0&h\\0&1&k\\0&0&1\end{array}\right] .\left[\begin{array}{ccc}cosx&-sinx&0\\sinx&cosx&0\\0&0&1\end{array}\right] .\left[\begin{array}{ccc}1&0&-h\\0&1&-k\\0&0&1\end{array}\right]$

Now,

New obj matrix = T x obj matrix

                          = $\left[\begin{array}{ccc}0&1&5\\0&1&2\\1&1&1\end{array}\right]$

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Answer 2

Answer:

Our matrix becomes
$\left[\begin{array}{ccc}0&1&5\\0&1&2\\1&1&1\end{array}\right]$

Step-by-step explanation:

It is given to us that we have to form a 3X3 matrix by translating (3,1) by rotating it about origin through 45 degree

As we know that the representation of transverse axis on matrix is
$\left[\begin{array}{ccc}1&0&h\\0&1&k\\0&0&1\end{array}\right] \\\\\\\left[\begin{array}{ccc}1&0&3\\0&1&1\\0&0&1\end{array}\right]$

Now for rotation we have
$\left[\begin{array}{ccc}cosx&-sinx&0\\sinx&cos x&0\\0&0&1\end{array}\right] \\\\\\\\\left[\begin{array}{ccc}\frac{1}{\sqrt{2} }&-\frac{1}{\sqrt{2} } &3\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} } &0\\0&0&1\end{array}\right]$

So the final 3X3 matrix becomes
$\left[\begin{array}{ccc}1&0&3\\0&1&1\\0&0&1\end{array}\right] X\left[\begin{array}{ccc}\frac{1}{\sqrt{2} }&-\frac{1}{\sqrt{2} } &3\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} } &0\\0&0&1\end{array}\right] X\left[\begin{array}{ccc}1&0&3\\0&1&1\\0&0&1\end{array}\right] = \left[\begin{array}{ccc}0&1&5\\0&1&2\\1&1&1\end{array}\right]$

Hence our matrix becomes
$\left[\begin{array}{ccc}0&1&5\\0&1&2\\1&1&1\end{array}\right]$
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