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

When a point $(x, y)$ in the $xy$-coordinate system is rotated $90^\circ$ to get to the $x'y'$-coordinate system, the new coordinates become $(-y, x)$. For the point $(2, 2)$, this results in $(-2, 2)$, but due to a mistake in my explanation, the correct transformation should actually result in the coordinates being flipped and one of them negated, leading to $(y, -x)$, which correctly for $(2, 2)$ would be $(2, -2)$, aligning with choice C. My initial explanation mistakenly applied the transformation incorrectly. The correct transformation for a $90^\circ$ rotation should indeed result in $(2, -2)$, aligning with the principles of rotation in the coordinate system.

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