The point P (−1,2)
is rotated to become P′ (2,1)
. Describe the rotation by degree and direction.
To describe the rotation of point P(-1,2) to P'(2,1), we can first find the angle of rotation using the coordinates of the vectors OP and OP'.
Let's consider the vectors OP and OP' as vectors u and v, respectively, where O is the origin.
The vector u is u = (-1,2) and the vector v is v = (2,1).
To find the angle between these vectors, we can use the dot product formula:
u · v = |u| * |v| * cos(theta)
where |u| and |v| are the magnitudes of the vectors and theta is the angle between them.
The magnitude of vector u is |u| = sqrt((-1)^2 + 2^2) = sqrt(1+4) = sqrt(5).
The magnitude of vector v is |v| = sqrt(2^2 + 1^2) = sqrt(4+1) = sqrt(5).
The dot product of u and v is u · v = (-1)*(2) + (2)*(1) = -2 + 2 = 0.
Using the dot product formula, we can solve for cos(theta):
0 = sqrt(5) * sqrt(5) * cos(theta)
cos(theta) = 0 / (sqrt(5) * sqrt(5))
cos(theta) = 0
From this, we see that the angle theta is 90 degrees or π/2 radians.
Therefore, the rotation from P(-1,2) to P'(2,1) is a rotation of 90 degrees in a counter-clockwise direction.
To describe the rotation of point P (-1,2) to P' (2,1), we need to find the degree and direction of the rotation.
To do that, we can use the distance and direction between the two points.
First, let's calculate the change in x and y coordinates:
Δx = x' - x = 2 - (-1) = 3
Δy = y' - y = 1 - 2 = -1
Now, we can calculate the angle of rotation using the arctangent function:
θ = arctan(Δy/Δx) = arctan(-1/3)
Using a calculator, arctan(-1/3) is approximately -18.43°.
Since the rotation is from P to P', which means it goes counterclockwise, we can determine that the rotation is in the negative direction.
Therefore, the rotation from P to P' is approximately -18.43° counterclockwise.
To describe the rotation of point P(-1, 2) to point P'(2, 1), we can determine the number of degrees and the direction of the rotation using the following steps:
Step 1: Find the translation vector from the original point to the rotated point.
The translation vector is determined by subtracting the coordinates of the original point from the coordinates of the rotated point:
Translation Vector (Δx, Δy) = (2 - (-1), 1 - 2) = (3, -1)
Step 2: Determine the distance and direction of the translation vector.
The distance of the translation vector can be calculated using the Pythagorean theorem:
Distance = √(Δx^2 + Δy^2) = √(3^2 + (-1)^2) = √(9 + 1) = √10.
The direction of the translation vector can be determined using the arctangent function:
Angle = arctan(Δy/Δx) = arctan(-1/3) = -18.43 degrees (rounded to two decimal places).
Step 3: Convert the direction of the translation vector to describe the rotation.
Since a rotation by 180 degrees is equivalent to a translation by the same vector in the opposite direction, we can express the rotation of point P to point P' as a rotation of 180 - 18.43 degrees = 161.57 degrees in the opposite direction.
Therefore, the rotation of point P(-1, 2) to point P'(2, 1) is a rotation of approximately 161.57 degrees in the counterclockwise direction.