Transformasi T mewakili dilatasi [O,2], dilanjutkan refleksi terhadap garus y = x, dan dilanjutkan r.... Question from @CintyaNufus

May 2019 1 278 Report

Transformasi T mewakili dilatasi [O,2], dilanjutkan refleksi terhadap garus y = x, dan dilanjutkan rotasu R[O, -1/2π].
a. Nyatakan T sebagai transformasi tunggal.
b. Tentukan koordinat titik A jika bayangannya oleh transformasi T adalah A'(12, -8).

To9ha

Verified answer

Jawab
T1 = D[O,2]
$T_1\,=\, \left[\begin{array}{ccc}2&0\\0&2\end{array}\right]$

T2 = R(y = x)
$T_2\,=\, \left[\begin{array}{ccc}0&1\\1&0\end{array}\right]$

T3 = R[ O, -1/2π] = R[O, -90]
$T_3\,=\,
 
\left[\begin{array}{ccc}cos\,(-90)&-sin\,(-90)\\-sin\,(-90)&cos\,(-90)\end{array}\right]\,=\,
 \left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]$

a) Matriks transformasi tunggal T4 = T3 o T2 o T1
$T_4\,=\,
 
\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]\,\left[\begin{array}{ccc}0&1\\1&0\end{array}\right]\,\left[\begin{array}{ccc}2&0\\0&2\end{array}\right]
$

$T_4\,=\,\left[\begin{array}{ccc}2&0\\0&-2\end{array}\right]$

b) [T4] [A] = [A']
A = [T4]⁻¹ [ A ' ]

$A\,=\,
 \frac{1}{2(-2)-0} 
\left[\begin{array}{ccc}-2&0\\0&2\end{array}\right]\, 
\left[\begin{array}{ccc}12\\-8\end{array}\right]$

$A\,=\, \frac{1}{-4} \, \left[\begin{array}{ccc}24\\-16\end{array}\right]$

A = ( 6, 4 )