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

Answer 1

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).