A port to three is first rotated 90° counterclockwise about the origin and then reflected over the X axis. What are the coordinates for the final point
To rotate a point 90° counterclockwise about the origin, we can use the following rotation matrix:
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]
Since we want to rotate 90° counterclockwise, we have θ = -90° = -π/2 radians. Therefore, the rotation matrix becomes:
[ cos(-π/2) -sin(-π/2) ]
[ sin(-π/2) cos(-π/2) ]
Simplifying:
[ 0 1 ]
[ -1 0 ]
Now, let's apply this rotation matrix to the original point (3,3):
[ 0 1 ] [ 3 ]
[ -1 0 ] * [ 3 ] = [ 3*(-1) + 3*0, 3*0 + 3*1 ] = [ -3, 3 ]
So, after the 90° counterclockwise rotation, the point (3,3) becomes (-3,3).
Now, let's reflect this rotated point over the X-axis. Reflecting a point over the X-axis simply changes the sign of the y-coordinate.
Therefore, the final point is (-3, -3).
To find the coordinates for the final point after a port to three is first rotated 90° counterclockwise about the origin and then reflected over the X-axis, we can follow these steps:
Step 1: Start with the coordinates of the original point: (x, y).
Step 2: Rotate the point 90° counterclockwise about the origin. This rotation can be achieved by swapping the x and y coordinates and changing the sign of the new x coordinate. So the new coordinates after rotation will be: (-y, x).
Step 3: Reflect the rotated point over the X-axis. This can be achieved by changing the sign of the y-coordinate. So the final coordinates after reflection will be: (-y, -x).
Therefore, the coordinates for the final point are (-y, -x).