In the xy - plane, line l has equation y= -x. Point P lies on l and has coordinates (-2,2). If l is rotated counterclockwise 45 degrees about the origin, what will be the coordinates of the image of P under this rotation.
Recall your rotation matrix.
(x,y) -> (x',y') where
x' = xcosθ - ysinθ
y' = xsinθ + ycosθ
So, plug in your θ.
You can easily check your work by noting that the point will rotate down to the x-axis.
To find the coordinates of the image of point P after rotating counterclockwise 45 degrees about the origin, you can use the rotation matrix.
The rotation matrix for a counterclockwise rotation of angle θ about the origin is:
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]
In this case, the angle of rotation is 45 degrees.
Let's start by substituting the given coordinates of point P into the rotation matrix:
[ cos(45) -sin(45) ]
[ sin(45) cos(45) ]
The rotation matrix becomes:
[ √2/2 -√2/2 ]
[ √2/2 √2/2 ]
Now, multiply the rotation matrix by the coordinates of point P:
[ √2/2 -√2/2 ] [ -2 ]
[ √2/2 √2/2 ] * [ 2 ]
This gives us:
[ (√2/2)(-2) + (-√2/2)(2) ]
[ (√2/2)(2) + (√2/2)(2) ]
Simplifying the equation, we get:
[ -√2 - √2 ]
[ √2 + √2 ]
So the coordinates of the image of point P after rotating counterclockwise 45 degrees about the origin are (-2√2, 2√2).