Identify the point that applies a rotation function mapping for P in a 270-degree counterclockwise rotation about the origin.
The point that applies a rotation function mapping for P in a 270-degree counterclockwise rotation about the origin is the origin itself (0, 0).
To apply a 270-degree counterclockwise rotation to a point P about the origin, we can use the following rotation matrix:
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]
For a 270-degree counterclockwise rotation, θ is equal to (270 degrees) converted to radians = (3π/2) radians.
So, the rotation matrix for a 270-degree counterclockwise rotation about the origin is:
[ cos(3π/2) -sin(3π/2) ]
[ sin(3π/2) cos(3π/2) ]
Simplifying,
[ cos(3π/2) -sin(3π/2) ]
[ sin(3π/2) cos(3π/2) ]
= [ 0 -1 ]
[ 1 0 ]
Therefore, the rotation function mapping for P is:
P' = [ cos(3π/2) -sin(3π/2) ] * P
[ sin(3π/2) cos(3π/2) ]
To identify the point that applies a 270-degree counterclockwise rotation about the origin, we will use a basic understanding of transformations in the Cartesian coordinate system.
First, let's assume the original point P has the coordinates (x, y). To perform a counterclockwise rotation of 270 degrees about the origin, we need to apply the following formulas:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, θ represents the angle of rotation, which is 270 degrees. However, since these formulas use radians, we need to convert 270 degrees to radians:
θ = (270 * π) / 180 = 3π / 2
Now, substitute the values into the formulas:
x' = x*cos(3π/2) - y*sin(3π/2)
y' = x*sin(3π/2) + y*cos(3π/2)
Simplifying these expressions, we have:
x' = -y
y' = x
So, the rotated point P' is represented by the coordinates (-y, x).