Given a point ( x, y ) what is the new point when the point is rotate counterclockwise about the origin 270 degrees?...
To rotate a point counterclockwise about the origin, we use the rotation matrix:
[x'] [ cos(θ) -sin(θ) ] [x]
[y'] = [ sin(θ) cos(θ) ] * [y]
Since θ = 270 degrees, we can substitute the values:
[x'] [ cos(270°) -sin(270°) ] [x]
[y'] = [ sin(270°) cos(270°) ] * [y]
Simplifying the trigonometric functions:
[x'] [ 0 -1 ] [x]
[y'] = [ 1 0 ] * [y]
Multiplying the matrices:
x' = 0*x - 1*y = -y
y' = 1*x + 0*y = x
Therefore, the new point (x', y') when the point (x, y) is rotated counterclockwise about the origin 270 degrees is (-y, x).
given a point on a geometric figure (1, 2), what is the new point when the point is rotated clockwise about the origin 90 degrees?
To rotate a point clockwise about the origin, we use the rotation matrix:
[x'] [ cos(θ) sin(θ) ] [x]
[y'] = [-sin(θ) cos(θ) ] * [y]
Since θ = 90 degrees, we can substitute the values:
[x'] [ cos(90°) sin(90°) ] [x]
[y'] = [-sin(90°) cos(90°) ] * [y]
Simplifying the trigonometric functions:
[x'] [ 0 1 ] [x]
[y'] = [-1 0 ] * [y]
Multiplying the matrices:
x' = 0*x + 1*y = y
y' = -1*x + 0*y = -x
Therefore, the new point (x', y') when the point (1, 2) is rotated clockwise about the origin 90 degrees is (2, -1).
given a point on a line segment (-4, 50, what is the new point when the point is rotated clockwise about the origin 270 degrees?
To rotate a point clockwise about the origin, we use the rotation matrix:
[x'] [ cos(θ) sin(θ) ] [x]
[y'] = [-sin(θ) cos(θ) ] * [y]
Since θ = 270 degrees, we can substitute the values:
[x'] [ cos(270°) sin(270°) ] [x]
[y'] = [-sin(270°) cos(270°) ] * [y]
Simplifying the trigonometric functions:
[x'] [ 0 -1 ] [x]
[y'] = [ 1 0 ] * [y]
Multiplying the matrices:
x' = 0*x - 1*y = -y
y' = 1*x + 0*y = x
Therefore, the new point (x', y') when the point (-4, 50) is rotated clockwise about the origin 270 degrees is (-50, -4).
given a point on a geometric figure (x, y), what is the new point when the point is rotated clockwise about the origin 270 degrees?
To rotate a point clockwise about the origin, we use the rotation matrix:
[x'] [ cos(θ) sin(θ) ] [x]
[y'] = [-sin(θ) cos(θ) ] * [y]
Since we want to rotate the point 270 degrees clockwise, we substitute θ = -270 degrees (negative because it is clockwise):
[x'] [ cos(-270°) sin(-270°) ] [x]
[y'] = [-sin(-270°) cos(-270°) ] * [y]
Simplifying the trigonometric functions:
[x'] [ cos(270°) -sin(270°) ] [x]
[y'] = [ sin(270°) cos(270°) ] * [y]
Since cos(270°) = 0 and sin(270°) = -1, we can substitute these values:
[x'] [ 0 -1 ] [x]
[y'] = [ 1 0 ] * [y]
Multiplying the matrices:
x' = 0*x - 1*y = -y
y' = 1*x + 0*y = x
Therefore, the new point (x', y') when the point (x, y) is rotated clockwise about the origin 270 degrees is (-y, x).
given a point (1, 2) on a geometric figure, what is the new point when the point is rotated counterclockwise about the origin 90 degrees?
To rotate a point counterclockwise about the origin, we use the rotation matrix:
[x'] [ cos(θ) -sin(θ) ] [x]
[y'] = [ sin(θ) cos(θ) ] * [y]
Since we want to rotate the point (1, 2) counterclockwise about the origin 90 degrees, we substitute θ = 90 degrees:
[x'] [ cos(90°) -sin(90°) ] [1]
[y'] = [ sin(90°) cos(90°) ] * [2]
Simplifying the trigonometric functions:
[x'] [ 0 -1 ] [1]
[y'] = [ 1 0 ] * [2]
Multiplying the matrices:
x' = 0*1 - 1*2 = -2
y' = 1*1 + 0*2 = 1
Therefore, the new point when the point (1, 2) is rotated counterclockwise about the origin 90 degrees is (-2, 1).