We are given that there is a counter clockwise rotation about the origin.
This means that the original x axis becomes the new y axis and the original y axis becomes the new x axis. But from the fact that (2,0) is taken to (0,2) and (0,3) is taken to (-3,0), we see that the new x axis now points in the direction of the positive y direction but the new y axis points in the direction of the negative x axis. For any point with original coordinates (a, b) the new coordinates are (-b, a).
Therefore the point (4, 1) is taken to (-1, 4).
90 degree about the origin (0,0) takes (2,0) to (0,2)
90 degree about the origin takes (0,3) = (-3, 0)
To find where does the rotation of (4,1) about the origin goes to.
Let A bethe point (4,1). and O is the origin (0 , 0).
Then the slope of OA = 4/1
Length of OA = sqrt(4^2+1) = sqr17.
When A is rotated by 90 degree , it moves to a place B in 2nd quadrant B (x,y).
OB = OA = sqrt17.
x^2+y^2 = 17...(1)
Then the slope of B = y/x
Product of the slope of OB andOA = -1 as the angle AOB = 90 degrees.
(y/x)*4 = -1
Or y = -4x.........(2).
Put y = -4x in (1):
x^2+(-4x)^2 = 17
17x^2 = 1
x ^2 = 1.
x = sqrt1.
x = 1 or x= -1.
When x = 1, y = -4x = -4*1 = -4, gives the position of rotation of 90 degree clockwise direction.
When x = -1 , y = -(-1) = 4. Or
B(x, y) = B ( -1 , 4) is the position of A(1,4) after 90 degree anti clockwise rotation.