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

Answer 1

To determine the rotation that transforms the point \( P(-1, 2) \) to the point \( P'(2, 1) \), we can follow these steps:

1. Find the center of rotation: Since we need to describe the rotation specifically, we will assume a rotation around the origin \( (0, 0) \) first. We can find the angle of rotation using the properties of the points involved.

2. Use vectors:

   • The initial vector from the origin to point \( P \) is \( \vec{OP} = (-1, 2) \).
   • The resulting vector from the origin to point \( P' \) is \( \vec{OP'} = (2, 1) \).

3. Find the angle of rotation: We need to calculate the angle between the two vectors. We can use the dot product to find the angle \( \theta \): \[ \vec{OP} \cdot \vec{OP'} = (-1)(2) + (2)(1) = -2 + 2 = 0 \] The dot product is zero, indicating that the two vectors are perpendicular.

4. Determine the rotation direction:

   • We can find out in which direction \( P \) has rotated to point \( P'\). If we visualize or sketch the vectors, we see that the vector pointing to \( P' \) can be obtained by rotating the vector pointing to \( P \) by \( 90^\circ \) clockwise.

Therefore, the rotation that transforms point \( P(-1, 2) \) to point \( P'(2, 1) \) is a 90-degree rotation clockwise (or -90 degrees) around the origin.