Use [tex]\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right][/tex] to rotate [tex]\left[\begin{array}{c}12 \\ -7\end{array}\right] 90^{\circ}[/tex] clockwise. Identify the transformed vector.



Answer :

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].