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[tex]\left[\begin{array}{l}9 \\ 8\end{array}\right][/tex] was transformed to [tex]\left[\begin{array}{c}8 \\ -9\end{array}\right][/tex] using [tex]\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right][/tex]. Identify the transformation.

A. [tex]\left[\begin{array}{l}9 \\ 8\end{array}\right][/tex] was reflected over the [tex]x[/tex]-axis.

B. [tex]\left[\begin{array}{l}9 \\ 8\end{array}\right][/tex] was rotated [tex]90^{\circ}[/tex] counterclockwise.

C. [tex]\left[\begin{array}{l}9 \\ 8\end{array}\right][/tex] was rotated [tex]90^{\circ}[/tex] clockwise.

Sagot :

GinnyAnswer
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]
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