give a point (x, y), what is the new point when the point is rotated counterclockwise about the origin 270 degrees

Answer 1

To find the new point when a point (x, y) is rotated counterclockwise about the origin by 270 degrees, we need to apply a rotation matrix.

The rotation matrix for a counterclockwise rotation by θ degrees is:
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]

In this case, θ = 270 degrees.

Applying the rotation matrix:

[ cos(270) -sin(270) ] [ x ] = [ cos(270)x - sin(270)y ]
[ sin(270) cos(270) ] [ y ] [ sin(270)x + cos(270)y ]

Now, cos(270) = 0 and sin(270) = -1, so the transformation becomes:

[ 0 -(-1) ] [ x ] = [ 0x - (-1)y ]
[ -1 0 ] [ y ] [ -y 0x ]

Simplifying further:

[ 0 1 ] [ x ] = [ y ]
[ -1 0 ] [ y ] [ -x ]

Therefore, the new point after rotating (x, y) counterclockwise by 270 degrees about the origin is (-y, x).

Answer 2

thats wrong

Answer 3

Apologies for the mistake. Let's correct it.

To find the new point when a point (x, y) is rotated counterclockwise about the origin by 270 degrees, we can use the fact that a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation.

The rotation matrix for a clockwise rotation by θ degrees is:
[ cos(θ) sin(θ) ]
[ -sin(θ) cos(θ) ]

In this case, θ = 90 degrees.

Applying the rotation matrix:

[ cos(90) sin(90) ] [ x ] = [ cos(90)x + sin(90)y ]
[ -sin(90) cos(90) ] [ y ] [ -sin(90)x + cos(90)y ]

Now, cos(90) = 0 and sin(90) = 1, so the transformation becomes:

[ 0 1 ] [ x ] = [ 0x + 1y ]
[ -1 0 ] [ y ] [ -1x + 0y ]

Simplifying further:

[ 0 1 ] [ x ] = [ y ]
[ -1 0 ] [ y ] [ -x ]

Therefore, the new point after rotating (x, y) counterclockwise by 270 degrees about the origin is (y, -x).