Which of the following is the rule for rotating the point with coordinates $(x, y)$, $90^\circ$ counterclockwise about the origin?
A. $(x, y) \rightarrow(y,-x)$
B. $(x, y) \rightarrow(-y, x)$
C. $(x, y) \rightarrow(y, x)$
D. $(x, y) \rightarrow(x,-y)$
Answer (2)
The problem requires finding the transformation rule for a 90-degree counterclockwise rotation about the origin.
The transformation rule is derived using trigonometric relationships and the properties of rotation.
The correct transformation rule is ( x , y ) → ( − y , x ) .
The final answer is B .
Explanation
Problem Analysis Let's analyze the problem. We need to find the rule for rotating a point ( x , y ) 90 degrees counterclockwise about the origin. This means we need to determine how the coordinates x and y change after this rotation.
Rotation Rule Consider a point ( x , y ) in the Cartesian plane. To rotate this point 90 degrees counterclockwise about the origin, we can use the following transformation: ( x , y ) → ( − y , x ) This means the new x -coordinate will be the negative of the original y -coordinate, and the new y -coordinate will be the original x -coordinate.
Verification with Examples Let's check this with an example. Suppose we have the point ( 1 , 0 ) . After a 90-degree counterclockwise rotation, it should become ( 0 , 1 ) . Using our rule, ( 1 , 0 ) → ( − 0 , 1 ) = ( 0 , 1 ) , which is correct.
Another example: the point ( 0 , 1 ) should become ( − 1 , 0 ) . Using our rule, ( 0 , 1 ) → ( − 1 , 0 ) , which is also correct.
Identifying the Correct Option Comparing our result with the given options: A. ( x , y ) → ( y , − x ) B. ( x , y ) → ( − y , x ) C. ( x , y ) → ( y , x ) D. ( x , y ) → ( x , − y )
The correct rule is option B.
Final Answer Therefore, the rule for rotating the point with coordinates ( x , y ) 90 degrees counterclockwise about the origin is: ( x , y ) → ( − y , x )
The final answer is B.
Examples
Imagine you're designing a simple game where a character needs to rotate 90 degrees counterclockwise. If the character's current position is (2, 3) on the game map, applying this rotation rule (-y, x) would move the character to (-3, 2). This ensures the character turns correctly in the game world, allowing for accurate navigation and interaction.
The rule for rotating the point ( x , y ) 90 degrees counterclockwise about the origin is given by ( x , y ) → ( − y , x ) . This correctly transforms the coordinates as shown through verification with examples. Thus, the chosen option is B.
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