An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Reflection in y-axis: ( x , y ) → ( − x , y ) , resulting in X ′ ( − 2 , 9 ) , Y ′ ( − 5 , 9 ) , Z ′ ( − 2 , 4 ) .
Anticlockwise rotation of 90 degrees: ( x , y ) → ( − y , x ) , resulting in X 2 ( − 9 , 2 ) , Y 2 ( − 9 , 5 ) , Z 2 ( − 4 , 2 ) .
Enlargement with scale factor -1: ( x , y ) → ( − x , − y ) , resulting in X 3 ( − 2 , − 9 ) , Y 3 ( − 5 , − 9 ) , Z 3 ( − 2 , − 4 ) .
The transformed coordinates are X ′ ( − 2 , 9 ) , Y ′ ( − 5 , 9 ) , Z ′ ( − 2 , 4 ) , X 2 ( − 9 , 2 ) , Y 2 ( − 9 , 5 ) , Z 2 ( − 4 , 2 ) , X 3 ( − 2 , − 9 ) , Y 3 ( − 5 , − 9 ) , Z 3 ( − 2 , − 4 ) .
Explanation
Problem Analysis The problem requires us to perform three geometric transformations on a triangle Δ X Y Z with vertices X ( 2 , 9 ) , Y ( 5 , 9 ) , Z ( 2 , 4 ) . The transformations are:
Reflection in the y-axis.
Anticlockwise rotation of 90 degrees about the origin.
Enlargement from the origin with a scale factor of -1.
We need to find the coordinates of the vertices of the transformed triangles after each transformation.
Applying Transformations
Reflection in the y-axis:
When a point ( x , y ) is reflected in the y-axis, its image is ( − x , y ) . Therefore:
X ( 2 , 9 ) → X ′ ( − 2 , 9 )
Y ( 5 , 9 ) → Y ′ ( − 5 , 9 )
Z ( 2 , 4 ) → Z ′ ( − 2 , 4 )
Anticlockwise rotation of 90 degrees about the origin:
When a point ( x , y ) is rotated 90 degrees anticlockwise about the origin, its image is ( − y , x ) . Therefore:
X ( 2 , 9 ) → X 2 ( − 9 , 2 )
Y ( 5 , 9 ) → Y 2 ( − 9 , 5 )
Z ( 2 , 4 ) → Z 2 ( − 4 , 2 )
Enlargement from the origin with a scale factor of -1:
When a point ( x , y ) is enlarged from the origin with a scale factor of -1, its image is ( − x , − y ) . Therefore:
X ( 2 , 9 ) → X 3 ( − 2 , − 9 )
Y ( 5 , 9 ) → Y 3 ( − 5 , − 9 )
Z ( 2 , 4 ) → Z 3 ( − 2 , − 4 )
Final Coordinates The coordinates of the transformed triangles are:
Δ X ′ Y ′ Z ′ : X ′ ( − 2 , 9 ) , Y ′ ( − 5 , 9 ) , Z ′ ( − 2 , 4 )
Δ X 2 Y 2 Z 2 : X 2 ( − 9 , 2 ) , Y 2 ( − 9 , 5 ) , Z 2 ( − 4 , 2 )
Δ X 3 Y 3 Z 3 : X 3 ( − 2 , − 9 ) , Y 3 ( − 5 , − 9 ) , Z 3 ( − 2 , − 4 )
Examples
Geometric transformations are used extensively in computer graphics, animation, and game development. For instance, when creating a 2D game, developers use transformations like reflection, rotation, and scaling to manipulate the position and orientation of game objects, such as characters or obstacles. These transformations allow for dynamic and interactive gameplay, where objects can move, rotate, and change size in response to player actions or game events. Understanding these transformations is crucial for creating visually appealing and engaging gaming experiences.
