The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system. $\theta = 90 ^ { \circ }$ , $( 2,2 )$
A) $$( - 2,0 )$$
B) $( 0,3 )$
C) $( 2 , - 2 )$
D) $( 2,0 )$
E) $( - 2,2 )$

Question

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.​ $\theta = 90 ^ { \circ }$ , $( 2,2 )$ ​
A) $$( - 2,0 )$$
B) $( 0,3 )$
C) $( 2 , - 2 )$
D) $( 2,0 )$
E) $( - 2,2 )$

Quizplus · Accepted Answer

To rotate a point by a certain angle, we use the following transformation matrix:

| cos(theta) -sin(theta) | 
| sin(theta)  cos(theta) |

where theta is the angle of rotation. In this case, the angle of rotation is 11ea99db_b805_16c3_94ac_ad0df2f16dec_TB4590_11 degrees. Converting this angle to radians gives:

11ea99db_b805_16c3_94ac_ad0df2f16dec_TB4590_11 * pi / 180 = 0.20000000298 pi

Using this angle, we can form the transformation matrix:

| cos(0.20000000298 pi) -sin(0.20000000298 pi) |
| sin(0.20000000298 pi)  cos(0.20000000298 pi) |

We can then multiply this matrix with the coordinates of the point in the 11ea99db_b805_3dd5_94ac_5bc1667de2e5_TB4590_11 -coordinate system to obtain the coordinates of the point in the rotated coordinate system:

Therefore, the answer is choice C.

The x′y′x ^ { \prime } y ^ { \prime }x′y′

The $x ^ { \prime } y ^ { \prime }$