Give a point on a geometric figure (x,y), what is the new point when the figure is rotated clockwise about the origin 270°?
To find the new coordinates after rotating a point clockwise about the origin, we can use the rotation matrix formula.
The formula to rotate a point (x, y) clockwise about the origin by an angle θ is:
x' = x * cos(θ) + y * sin(θ)
y' = -x * sin(θ) + y * cos(θ)
In this case, the angle of rotation is 270°. Converting this to radians (since trigonometric functions in most programming languages use radians), we have:
θ = 270° * (π/180°) = 3π/2 radians.
Let's assume the given point is (x, y). Plugging this into the rotation matrix formula, we get:
x' = x * cos(3π/2) + y * sin(3π/2)
y' = -x * sin(3π/2) + y * cos(3π/2)
Now, let's simplify these equations:
cos(3π/2) = 0
sin(3π/2) = -1
cos(3π/2) = 0
sin(3π/2) = -1
x' = x * 0 + y * (-1)
y' = -x * (-1) + y * 0
Therefore, the new coordinates (x', y') after rotating the point (x, y) clockwise about the origin by 270° are:
x' = -y
y' = x