The vector has components +5 and +7 along the x and y axes respectively. If the vector is now rotated 90 degrees counterclockwise relative to the original axes, the vector's components are now
A) −7; −5.
B) 7; −5.
C) −7; 5.
D) 7; 5.
E) 7; 0.

Question

The vector  has components +5 and +7 along the x and y axes respectively. If the vector is now rotated 90 degrees counterclockwise relative to the original axes, the vector's components are now
A) −7; −5. 
B) 7; −5. 
C) −7; 5. 
D) 7; 5. 
E) 7; 0.

Quizplus · Accepted Answer

We can use the rotation matrix formula to find the new components of the vector after it has been rotated counterclockwise by 90 degrees. The rotation matrix is given by:
$$\begin{bmatrix} \cos	heta & -\sin	heta\ \sin	heta & \cos	heta \end{bmatrix}$$
where $	heta=90^\circ$ in our case.
Thus, we have:
$$\begin{bmatrix} \cos 90^\circ & -\sin 90^\circ\ \sin 90^\circ & \cos 90^\circ \end{bmatrix}\begin{bmatrix} 5 \ 7 \end{bmatrix}=\begin{bmatrix} -7 \ 5 \end{bmatrix}$$
Therefore, the new components of the vector are -7 and 5 along the x and y axes, respectively. The answer is C.

The Vector Has Components +5 and +7 Along the X