The point p (-1,2) is rotated to become P' z(2,1). Describe the rotation by degree and direction

Answer 1

To determine the degree and direction of the rotation from point \( P(-1, 2) \) to point \( P'(2, 1) \), we can use the following steps:

1. Find the Center of Rotation: In this case, we need to assume a likely center for rotation. The most common center used in such problems is the origin (0, 0). However, since we do not know the actual center, we will assume the origin for this calculation.

2. Find the Original and New Coordinates:

   • Original point \( P(-1, 2) \)
   • New point \( P'(2, 1) \)

3. Calculate the Angles:

   • Convert the coordinates of both points into polar coordinates or calculate the angles with respect to the center of rotation (the origin).
   • For \( P(-1, 2) \):
     • \(\tan^{-1}\left(\frac{2}{-1}\right) = \tan^{-1}(-2) \approx 116.57^\circ\) (in the second quadrant)
   • For \( P'(2, 1) \):
     • \(\tan^{-1}\left(\frac{1}{2}\right) \approx 26.57^\circ\) (in the first quadrant)

4. Calculate the Difference in Angles:

   • To find the angle of rotation, subtract the original angle from the new angle: \[ \text{Rotation Angle} = 26.57^\circ - 116.57^\circ = -90^\circ \]
   • A negative angle indicates clockwise rotation.

5. Conclusion: The point \( P(-1, 2) \) is rotated to point \( P'(2, 1) \) by a rotation of 90 degrees clockwise around the origin.

If the center of rotation was other than the origin, the angles would change based on the center's coordinates, but without that information, we generally calculate it based on the origin.