An angle of rotation is specified, followed by the coordinates of a point in the $x ^ { \prime } - y ^ { \prime }$ system. Find the coordinates of the point with respect to the $x - y$ system. $\theta = 120 ^ { \circ }$ $\left( x ^ { \prime } , y ^ { \prime } \right) = ( \sqrt { 3 } , 8 )$
A) $\left( \frac { 5 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$
B) $\left( - \frac { 9 \sqrt { 3 } } { 2 } , - \frac { 5 } { 2 } \right)$
C) $\left( - \frac { 5 } { 2 } , - \frac { 9 \sqrt { 3 } } { 2 } \right)$
D) $\left( \frac { 9 \sqrt { 3 } } { 2 } , - \frac { 5 } { 2 } \right)$
E) $\left( \frac { 9 \sqrt { 3 } } { 2 } , \frac { 5 } { 2 } \right)$

Question

Quizplus · Accepted Answer

To find the coordinates of the point in the $x - y$ system, we will use the rotation matrix:

$$\begin{bmatrix} \cos \theta & - \sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} x \ y \end{bmatrix}$$

Plugging in the values we get:

$$\begin{bmatrix} \cos 120 ^{\circ} & - \sin 120 ^{\circ} \ \sin 120 ^{\circ} & \cos 120 ^{\circ} \end{bmatrix} \begin{bmatrix} \sqrt{3} \ 8 \end{bmatrix} = \begin{bmatrix} -\frac{9\sqrt{3}}{2} \ -\frac{5}{2} \end{bmatrix}$$

Therefore, the coordinates in the $x-y$ system are $\boxed{\textbf{(B)} \left(-\frac{9\sqrt{3}}{2}, -\frac{5}{2}\right)}$.

An Angle of Rotation Is Specified, Followed by the Coordinates x′−y′x ^ { \prime } - y ^ { \prime }x′−y′

An Angle of Rotation Is Specified, Followed by the Coordinates $x ^ { \prime } - y ^ { \prime }$