- Define the transformation matrix \(T\) and the vertex matrix \(R\).
- Multiply the transformation matrix by each vertex coordinate vector \(\begin{bmatrix} x \ y \end{bmatrix}\) to find the new points \((-x, y)\).
- Calculate the transformed vertices to be \((0,0), (0,3), (-3,3), (-3,0)\).
- Final Answer: \(\boxed{(0,0), (0,3), (-3,3), (-3,0)}\)
Explanation
- Understanding the Transformation
The vertices of the rectangle are represented as columns in the matrix \(R = \begin{bmatrix} 0 & 0 & 3 & 3 \ 0 & 3 & 3 & 0 \end{bmatrix}\). Each column vector \(\begin{bmatrix} x \ y \end{bmatrix}\) represents a point \((x, y)\) in the Cartesian plane. The rectangle is being transformed by the matrix \(T = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}\). To find the new coordinates, we need to multiply the transformation matrix \(T\) by the vertex matrix \(R\). This transformation represents a reflection across the y-axis, as the \(x\)-coordinates are negated while the \(y\)-coordinates remain unchanged.
- Computing the Transformed Coordinates
To compute the matrix product \(T \times R\), we perform the multiplication for each column:
\[T \times R = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 0 & 0 & 3 & 3 \ 0 & 3 & 3 & 0 \end{bmatrix}\]
For each column vector \(\begin{bmatrix} x_i \ y_i \end{bmatrix}\) in \(R\), the transformed vector is \(\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} x_i \ y_i \end{bmatrix} = \begin{bmatrix} -x_i \ y_i \end{bmatrix}\). Applying this to all columns:
- Column 1: \(\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\)
- Column 2: \(\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \ 3 \end{bmatrix} = \begin{bmatrix} 0 \ 3 \end{bmatrix}\)
- Column 3: \(\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \ 3 \end{bmatrix} = \begin{bmatrix} -3 \ 3 \end{bmatrix}\)
- Column 4: \(\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \ 0 \end{bmatrix} = \begin{bmatrix} -3 \ 0 \end{bmatrix}\)
- Concluding the Result
The resulting matrix is \(\begin{bmatrix} 0 & 0 & -3 & -3 \ 0 & 3 & 3 & 0 \end{bmatrix}\). The columns of this matrix correspond to the points \((0,0), (0,3), (-3,3), (-3,0)\). Comparing these results with the given options, we find that these coordinates perfectly match the fourth option. Therefore, the transformed rectangle's vertices are \((0,0), (0,3), (-3,3), (-3,0)\).
Examples
In computer graphics, transformations like this are essential for animating objects. By multiplying the coordinates of an object's vertices by a transformation matrix, an animator can easily reflect, rotate, or scale a 3D model or a 2D sprite. For example, if you wanted to flip a character model to face the opposite direction, you would use a reflection matrix similar to the one in this problem, which negates the x-coordinate of every vertex in the character's wireframe model.