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Sagot :
To find the coordinates of the transformed rectangle, we need to perform a linear transformation on the given matrix [tex]\( R \)[/tex] using the transformation matrix provided. Below is a detailed, step-by-step solution to determine the transformed coordinates.
Step 1: Identify the matrix [tex]\( R \)[/tex] which contains the vertices of the rectangle.
The given matrix [tex]\( R \)[/tex] is:
[tex]\[ R = \begin{bmatrix} 0 & 0 & 3 & 3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
Step 2: Identify the transformation matrix [tex]\( T \)[/tex].
The transformation matrix [tex]\( T \)[/tex] given is:
[tex]\[ T = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \][/tex]
Step 3: Perform the matrix multiplication of [tex]\( T \)[/tex] with [tex]\( R \)[/tex].
The multiplication [tex]\( T \times R \)[/tex] is performed as follows:
[tex]\[ T \times R = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & 0 & 3 & 3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
To compute each element of the resulting matrix, we take the dot product of the rows of [tex]\( T \)[/tex] with the columns of [tex]\( R \)[/tex].
- First column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 \\ 0 \end{bmatrix} = (-1 \times 0) + (0 \times 0) = 0 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 0 \end{bmatrix} = (0 \times 0) + (1 \times 0) = 0 \][/tex]
- Second column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 \\ 3 \end{bmatrix} = (-1 \times 0) + (0 \times 3) = 0 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 3 \end{bmatrix} = (0 \times 0) + (1 \times 3) = 3 \][/tex]
- Third column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 3 \\ 3 \end{bmatrix} = (-1 \times 3) + (0 \times 3) = -3 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 3 \end{bmatrix} = (0 \times 3) + (1 \times 3) = 3 \][/tex]
- Fourth column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 3 \\ 0 \end{bmatrix} = (-1 \times 3) + (0 \times 0) = -3 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 0 \end{bmatrix} = (0 \times 3) + (1 \times 0) = 0 \][/tex]
Thus, the resulting transformed matrix is:
[tex]\[ T \times R = \begin{bmatrix} 0 & 0 & -3 & -3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
Step 4: Extract the coordinates of the transformed rectangle.
By reading the columns of the transformed matrix, the coordinates of the transformed rectangle are:
[tex]\[ (0, 0), (0, 3), (-3, 3), (-3, 0) \][/tex]
Step 5: Match the result with the provided options.
The coordinates we found match with the option:
[tex]\[ (0,0), (0,3), (-3,3), (-3,0) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(0,0),(0,3),(-3,3),(-3,0)} \][/tex]
Step 1: Identify the matrix [tex]\( R \)[/tex] which contains the vertices of the rectangle.
The given matrix [tex]\( R \)[/tex] is:
[tex]\[ R = \begin{bmatrix} 0 & 0 & 3 & 3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
Step 2: Identify the transformation matrix [tex]\( T \)[/tex].
The transformation matrix [tex]\( T \)[/tex] given is:
[tex]\[ T = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \][/tex]
Step 3: Perform the matrix multiplication of [tex]\( T \)[/tex] with [tex]\( R \)[/tex].
The multiplication [tex]\( T \times R \)[/tex] is performed as follows:
[tex]\[ T \times R = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & 0 & 3 & 3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
To compute each element of the resulting matrix, we take the dot product of the rows of [tex]\( T \)[/tex] with the columns of [tex]\( R \)[/tex].
- First column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 \\ 0 \end{bmatrix} = (-1 \times 0) + (0 \times 0) = 0 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 0 \end{bmatrix} = (0 \times 0) + (1 \times 0) = 0 \][/tex]
- Second column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 \\ 3 \end{bmatrix} = (-1 \times 0) + (0 \times 3) = 0 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 3 \end{bmatrix} = (0 \times 0) + (1 \times 3) = 3 \][/tex]
- Third column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 3 \\ 3 \end{bmatrix} = (-1 \times 3) + (0 \times 3) = -3 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 3 \end{bmatrix} = (0 \times 3) + (1 \times 3) = 3 \][/tex]
- Fourth column:
[tex]\[ \begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} 3 \\ 0 \end{bmatrix} = (-1 \times 3) + (0 \times 0) = -3 \][/tex]
[tex]\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 0 \end{bmatrix} = (0 \times 3) + (1 \times 0) = 0 \][/tex]
Thus, the resulting transformed matrix is:
[tex]\[ T \times R = \begin{bmatrix} 0 & 0 & -3 & -3 \\ 0 & 3 & 3 & 0 \end{bmatrix} \][/tex]
Step 4: Extract the coordinates of the transformed rectangle.
By reading the columns of the transformed matrix, the coordinates of the transformed rectangle are:
[tex]\[ (0, 0), (0, 3), (-3, 3), (-3, 0) \][/tex]
Step 5: Match the result with the provided options.
The coordinates we found match with the option:
[tex]\[ (0,0), (0,3), (-3,3), (-3,0) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(0,0),(0,3),(-3,3),(-3,0)} \][/tex]
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