Answer :
Let's analyze and solve the problem step by step to identify the transformation applied to the vector.
Given:
- Initial vector: [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex]
- Transformed vector: [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex]
- Transformation matrix: [tex]\(\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\)[/tex]
### Step-by-Step Solution
1. Understand the Initial Setup:
We start with the initial vector [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex] and need to apply the given transformation matrix.
2. Apply the Transformation Matrix:
The transformation matrix [tex]\(\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\)[/tex] is used for the transformation. To apply the matrix transformation:
[tex]\[ \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] \left[\begin{array}{c}9 \\ 8\end{array}\right] = \left[\begin{array}{c}(0 \cdot 9 + 1 \cdot 8) \\ (-1 \cdot 9 + 0 \cdot 8)\end{array}\right] = \left[\begin{array}{c}8 \\ -9\end{array}\right] \][/tex]
3. Compare Result with Given Transformed Vector:
The resulting vector from the transformation is [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex], which matches the given transformed vector [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex].
4. Determine the Type of Transformation:
We know:
- A reflection over the [tex]\(x\)[/tex]-axis would change the sign of the second component only, leading to the vector [tex]\(\left[\begin{array}{c}9 \\ -8\end{array}\right]\)[/tex], which does not match our transformed vector.
- A [tex]\(90^\circ\)[/tex] counterclockwise rotation would result in [tex]\(\left[\begin{array}{c}-8 \\ 9\end{array}\right]\)[/tex], which also does not match.
- A [tex]\(90^\circ\)[/tex] clockwise rotation changes the vector as follows:
If you rotate the vector [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex] [tex]\(90^\circ\)[/tex] clockwise, you use the matrix:
[tex]\[ \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] \][/tex]
This matches our transformation matrix and the resulting vector [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex].
Therefore, the correct type of transformation applied to the vector is option C:
[tex]\[\boxed{\text{C. }\left[\begin{array}{c}9 \\ 8\end{array}\right]\ \text{was rotated}\ 90^\circ\ \text{clockwise}.}\][/tex]
Given:
- Initial vector: [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex]
- Transformed vector: [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex]
- Transformation matrix: [tex]\(\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\)[/tex]
### Step-by-Step Solution
1. Understand the Initial Setup:
We start with the initial vector [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex] and need to apply the given transformation matrix.
2. Apply the Transformation Matrix:
The transformation matrix [tex]\(\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\)[/tex] is used for the transformation. To apply the matrix transformation:
[tex]\[ \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] \left[\begin{array}{c}9 \\ 8\end{array}\right] = \left[\begin{array}{c}(0 \cdot 9 + 1 \cdot 8) \\ (-1 \cdot 9 + 0 \cdot 8)\end{array}\right] = \left[\begin{array}{c}8 \\ -9\end{array}\right] \][/tex]
3. Compare Result with Given Transformed Vector:
The resulting vector from the transformation is [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex], which matches the given transformed vector [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex].
4. Determine the Type of Transformation:
We know:
- A reflection over the [tex]\(x\)[/tex]-axis would change the sign of the second component only, leading to the vector [tex]\(\left[\begin{array}{c}9 \\ -8\end{array}\right]\)[/tex], which does not match our transformed vector.
- A [tex]\(90^\circ\)[/tex] counterclockwise rotation would result in [tex]\(\left[\begin{array}{c}-8 \\ 9\end{array}\right]\)[/tex], which also does not match.
- A [tex]\(90^\circ\)[/tex] clockwise rotation changes the vector as follows:
If you rotate the vector [tex]\(\left[\begin{array}{c}9 \\ 8\end{array}\right]\)[/tex] [tex]\(90^\circ\)[/tex] clockwise, you use the matrix:
[tex]\[ \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] \][/tex]
This matches our transformation matrix and the resulting vector [tex]\(\left[\begin{array}{c}8 \\ -9\end{array}\right]\)[/tex].
Therefore, the correct type of transformation applied to the vector is option C:
[tex]\[\boxed{\text{C. }\left[\begin{array}{c}9 \\ 8\end{array}\right]\ \text{was rotated}\ 90^\circ\ \text{clockwise}.}\][/tex]
