To solve for the transformed vector when rotating [tex]\(\left( \begin{array}{c} 12 \\ -7 \end{array} \right)\)[/tex] by [tex]\(90^\circ\)[/tex] clockwise using the rotation matrix [tex]\(\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)\)[/tex], follow these detailed steps:
1. Identify the given rotation matrix and the original vector:
- Rotation matrix: [tex]\(\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \)[/tex]
- Original vector: [tex]\(\left( \begin{array}{c} 12 \\ -7 \end{array} \right) \)[/tex]
2. Perform the matrix multiplication:
- The matrix multiplication involves taking the dot product of each row of the rotation matrix with the column vector.
Compute the first component of the transformed vector:
[tex]\[
\begin{array}{l}
0 \cdot 12 + 1 \cdot (-7) = 0 - 7 = -7
\end{array}
\][/tex]
Compute the second component of the transformed vector:
[tex]\[
\begin{array}{l}
-1 \cdot 12 + 0 \cdot (-7) = -12 + 0 = -12
\end{array}
\][/tex]
3. Combine the results to form the transformed vector:
[tex]\[
\left( \begin{array}{c} -7 \\ -12 \end{array} \right)
\][/tex]
Therefore, the vector [tex]\(\left( \begin{array}{c} 12 \\ -7 \end{array} \right)\)[/tex], when rotated [tex]\(90^\circ\)[/tex] clockwise, is transformed into the vector [tex]\(\left( \begin{array}{c} -7 \\ -12 \end{array} \right)\)[/tex].
