Determine the $\overline { X Y }$ - coordinates of $( - 2,5 )$ If the axes are rotated through an angle $\phi = 30 ^ { \circ }$
A) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { 2 + 5 \sqrt { 3 } } { 2 }$

B) $X = \frac { 2 \sqrt { 3 } } { 2 }$
, $Y = 1 - \frac { 5 \sqrt { 3 } } { 2 }$

C) $X = 5 - 2 \sqrt { 3 }$
, $Y = 1 + \frac { \sqrt { 3 } } { 2 }$

D) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { \sqrt { 3 } + 5 } { 2 }$

E) $X = \frac { 1 + 2 \sqrt { 3 } } { 2 }$
, $Y = - 5 + \frac { \sqrt { 3 } } { 2 }$

Question

Determine the $\overline { X Y }$ - coordinates of $( - 2,5 )$ If the axes are rotated through an angle $\phi = 30 ^ { \circ }$
A) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { 2 + 5 \sqrt { 3 } } { 2 }$

B) $X = \frac { 2 \sqrt { 3 } } { 2 }$
, $Y = 1 - \frac { 5 \sqrt { 3 } } { 2 }$

C) $X = 5 - 2 \sqrt { 3 }$
, $Y = 1 + \frac { \sqrt { 3 } } { 2 }$

D) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { \sqrt { 3 } + 5 } { 2 }$

E) $X = \frac { 1 + 2 \sqrt { 3 } } { 2 }$
, $Y = - 5 + \frac { \sqrt { 3 } } { 2 }$

Quizplus · Accepted Answer

To determine the $\overline {XY}$-coordinates of $(-2,5)$ after rotating the axes through an angle of 30 degrees, we need to use the following rotation matrix:

$$\begin{pmatrix} \cos \phi & -\sin \phi\ \sin \phi & \cos \phi \end{pmatrix} \begin{pmatrix} -2\ 5 \end{pmatrix}=\begin{pmatrix} X\ Y \end{pmatrix}$$

Substituting $\phi = 30^{\circ}$ and simplifying, we get:

$$\begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2}\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \begin{pmatrix} -2\ 5 \end{pmatrix}=\begin{pmatrix} X\ Y \end{pmatrix}$$

$$\begin{pmatrix} -\sqrt{3}-\frac{5}{2}\ 1+\frac{5\sqrt{3}}{2} \end{pmatrix}=\begin{pmatrix} X\ Y \end{pmatrix}$$

Therefore, the $\overline {XY}$-coordinates of $(-2,5)$ after rotating the axes through an angle of 30 degrees are $(X,Y) = \left(\frac{5-2\sqrt{3}}{2},\frac{2+5\sqrt{3}}{2}\right)$. The answer is choice letter A.

Determine The XY‾\overline { X Y }XY - Coordinates Of (−2,5)( - 2,5 )(−2,5)

Determine The $\overline { X Y }$ - Coordinates Of $( - 2,5 )$