The point L(a,b)in quadrant 1, is rotated 90 degrees about the origin and then reflected across the y-axis. What are the coordinates of the image of L?
rotating 90° about (0,0) takes
(x,y) -> (-y,x)
Reflection through x=0 takes (x,y) -> (-x,y)
so, (x,y) -> (-y,x) -> (y,x)
To find the coordinates of the image of point L(a, b) after rotating it 90 degrees counterclockwise about the origin, we can apply a 90-degree counterclockwise rotation matrix:
R = [0 -1]
[1 0]
The new coordinates after the rotation would be obtained by multiplying this rotation matrix by the original coordinates:
[x'] = [0 -1] [x]
[y'] [1 0] [y]
In this case, since point L is in quadrant 1, we know that a > 0 and b > 0. After applying the rotation matrix, the new coordinates are:
x' = 0*a + (-1)*b = -b
y' = 1*a + 0*b = a
So, after the 90-degree rotation, point L(a, b) in quadrant 1 will become (-b, a).
Now, to find the image of point L after reflecting it across the y-axis, we simply reverse the sign of the x-coordinate. Therefore, the coordinates of the image of L after the reflection would be (-(-b), a), which simplifies to (b, a).
Therefore, the coordinates of the image of L(a, b) in quadrant 1 after rotating it 90 degrees about the origin and then reflecting it across the y-axis are (b, a).