Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
-Rotate points through 45° and then scale the x-coordinate by 0.6 and the y-coordinate by 0.8.
A)
$\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \ 0.3 \sqrt { 2 } & 0.4 & 0 \ 0 & 0 & 1 \end{array} \right]$

B)

$\left[ \begin{array} { c c c } 0.3 \sqrt { 2 } & 0.3 \sqrt { 2 } & 0 \ - 0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \ 0 & 0 & 1 \end{array} \right]$

C)
$$\left[ \begin{array} { c c c }
0.3 \sqrt { 2 } & - 0.3 \sqrt { 2 } & 0 \
0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \
0 & 0 & 1
\end{array} \right]$$
D)
$$\left[ \begin{array} { c c c }
0 & - 0.6 & 0 \
0.8 & 0 & 0 \
0 & 0 & 1
\end{array} \right]$$

Question

Find the 3 × 3 matrix that produces the described composite 2D transformation, using homogeneous coordinates.
-Rotate points through 45° and then scale the x-coordinate by 0.6 and the y-coordinate by 0.8. 
A)
$\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \ 0.3 \sqrt { 2 } & 0.4 & 0 \ 0 & 0 & 1 \end{array} \right]$

B)

$\left[ \begin{array} { c c c } 0.3 \sqrt { 2 } & 0.3 \sqrt { 2 } & 0 \ - 0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \ 0 & 0 & 1 \end{array} \right]$

C)
$$\left[ \begin{array} { c c c } 
0.3 \sqrt { 2 } & - 0.3 \sqrt { 2 } & 0 \
0.4 \sqrt { 2 } & 0.4 \sqrt { 2 } & 0 \
0 & 0 & 1
\end{array} \right]$$
D)
$$\left[ \begin{array} { c c c } 
0 & - 0.6 & 0 \
0.8 & 0 & 0 \
0 & 0 & 1
\end{array} \right]$$

Quizplus · Accepted Answer

The correct matrix combines rotation by 45° and scaling of x by 0.6 and y by 0.8. The rotation matrix for 45° is $\left[ \begin{array} { c c } \cos(45°) & -\sin(45°) \ \sin(45°) & \cos(45°) \end{array} \right]$, which simplifies to $\left[ \begin{array} { c c } 0.707 & -0.707 \ 0.707 & 0.707 \end{array} \right]$ because $\cos(45°) = \sin(45°) = \frac{\sqrt{2}}{2}$. Scaling by 0.6 and 0.8 multiplies the x and y components by these values, respectively. Thus, the combined transformation matrix is $\left[ \begin{array} { c c c } 0.6 \cdot 0.707 & -0.6 \cdot 0.707 & 0 \ 0.8 \cdot 0.707 & 0.8 \cdot 0.707 & 0 \ 0 & 0 & 1 \end{array} \right]$, which simplifies to the matrix in choice C when considering $0.6 \cdot \frac{\sqrt{2}}{2} = 0.3\sqrt{2}$ and $0.8 \cdot \frac{\sqrt{2}}{2} = 0.4\sqrt{2}$.

Find the 3 × 3 Matrix That Produces the Described [0.3−0.4200.320.40001]\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \\ 0.3 \sqrt { 2 } & 0.4 & 0 \\ 0 & 0 & 1 \end{array} \right]​0.30.32​0​−0.42​0.40​001​​

Find the 3 × 3 Matrix That Produces the Described $\left[ \begin{array} { c c c } 0.3 & - 0.4 \sqrt { 2 } & 0 \\ 0.3 \sqrt { 2 } & 0.4 & 0 \\ 0 & 0 & 1 \end{array} \right]$