Sure, let's find the parametrization of the circle centered at the origin of radius 9 in the clockwise direction starting and ending at the point (0,9).
Firstly, let's start with the standard parametrization of a circle in the counterclockwise direction:
- The x-coordinate is r*cos(t) where r is the radius and t is the angle ranging from 0 to 2*pi.
- The y-coordinate is r*sin(t).
Now, we need to adjust this for the specifics of our problem: radius 9, starting at point (0,9) and moving in a clockwise direction.
1. The given circle is centered at the origin with a radius of r = 9.
2. To start at point (0,9) at t = 0, we shift our angle by pi/2 in the positive direction, because the point (0,9) is equivalent to pi/2 radians or 90 degrees in the standard unit circle in the positive direction.
3. For the clockwise direction, instead of increasing t from 0 to 2*pi, we will decrease t from 2*pi to 0.
So our final parameterization becomes:
- x = 9*cos(2*pi - t + pi/2)
- y = 9*sin(2*pi -t + pi/2)
So the position of any point on the circle in a clockwise direction can be obtained for any value of t in the interval [0, 2*pi]. When t = 0, we start at point (0,9) as required. As t increases from 0 to 2*pi, we move in the clockwise direction along the circle back to the point (0,9).