The point p (-1,2) is rotated to become (2,1). Describe the rotation by degree and direction.
1:-270degrees rotation
2:180 degrees rotation
3:-90 degrees rotation
4:90 degrees rotation
The rotation that takes point P(-1,2) to (2,1) can be described as a 90 degrees clockwise rotation.
To describe the rotation that transforms the point P(-1,2) to (2,1), we can use the following steps:
1. Find the translation vector from the origin to the original point P. The vector is given by the coordinates of P, which are (-1, 2).
2. Apply the rotation transformation.
1. For a 270-degree rotation, we rotate counterclockwise by 270 degrees. This rotation can also be described as a -90-degree rotation.
2. For a 180-degree rotation, we rotate counterclockwise by 180 degrees.
3. For a -90-degree rotation, we rotate clockwise by 90 degrees.
4. For a 90-degree rotation, we rotate counterclockwise by 90 degrees.
3. Apply the translation vector to the new coordinates.
Let's calculate the translations for each option:
1. -270 degrees rotation:
Translation vector: (-1, 2)
Rotate counterclockwise by -270 degrees:
New translation vector: (-1, 2)
Apply the translation vector to (2, 1):
New coordinates after applying the translation: (1, 3)
2. 180 degrees rotation:
Translation vector: (-1, 2)
Rotate counterclockwise by 180 degrees:
New translation vector: (1, -2)
Apply the translation vector to (2, 1):
New coordinates after applying the translation: (3, -1)
3. -90 degrees rotation:
Translation vector: (-1, 2)
Rotate clockwise by 90 degrees:
New translation vector: (2, 1)
Apply the translation vector to (2, 1):
New coordinates after applying the translation: (4, 2)
4. 90 degrees rotation:
Translation vector: (-1, 2)
Rotate counterclockwise by 90 degrees:
New translation vector: (-2, -1)
Apply the translation vector to (2, 1):
New coordinates after applying the translation: (0, 0)
Therefore, the rotation that transforms the point P(-1,2) to (2,1) is a 90-degree rotation counterclockwise OR a -270 degree rotation counterclockwise.
To describe the rotation of point P(-1, 2) to point Q(2, 1), we need to find the angle and direction of the rotation.
Step 1: Find the translation vector.
The translation vector is the difference between the two points:
Vector PQ = Q - P = (2, 1) - (-1, 2) = (3, -1)
Step 2: Find the angle.
To find the angle of the rotation, we can use the dot product formula:
cosθ = (Vector PQ • Vector P) / (||Vector PQ|| * ||Vector P||)
where • denotes the dot product and || denotes the magnitude (length) of a vector.
Vector P = (-1, 2)
Vector PQ = (3, -1)
||Vector P|| = √((-1)^2 + 2^2) = √5
||Vector PQ|| = √(3^2 + (-1)^2) = √10
cosθ = ((-1)(3) + (2)(-1)) / (√5 * √10) = (-3 - 2) / (√5 * √10) = -(5 / (√5 * √10)) = -√2 / 2
Using an inverse trigonometric function (arccos) to find the angle:
θ = arccos(-√2 / 2) ≈ 135 degrees
Step 3: Determine the direction.
To determine the direction of rotation, we can examine the signs of the translation vector's coordinates.
For a positive x-coordinate (3), the rotation is counterclockwise.
For a negative x-coordinate (-3), the rotation is clockwise.
Since the x-coordinate is positive, the rotation is counterclockwise.
Based on the calculations, the rotation from P(-1, 2) to Q(2, 1) is approximately a 135-degree counterclockwise rotation.