The point p (-1,2) is rotated to become (2,1). Describe the rotation by degree and direction.
To describe the rotation, we need to determine the angle and direction of rotation.
To find the angle, we can use the slope formula:
slope = (change in y-coordinate)/(change in x-coordinate)
Original slope:
m1 = (2 - 1)/(-1 - 2) = 1/3
After rotation, the slope must be the negative reciprocal of the original slope because perpendicular lines have negative reciprocal slopes. So, the new slope would be -3.
To find the angle, we can use the arctan function:
angle = arctan(slope)
Original angle:
θ1 = arctan(1/3) ≈ 18.43 degrees
After rotation, the new angle would be:
θ2 = arctan(-3)
However, since -3 can be in the second or fourth quadrant, we add or subtract 180 degrees to get the actual angle:
θ2 = arctan(-3) + 180 degrees ≈ 153.43 degrees or θ2 = arctan(-3) - 180 degrees ≈ -153.43 degrees
So, the rotation is either approximately 153.43 degrees counterclockwise or -153.43 degrees clockwise.
To determine the rotation, we can use the formula for rotating a point in the Euclidean plane. The formula is:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
where (x', y') are the coordinates of the resulting point, (x, y) are the coordinates of the initial point, and θ is the angle of rotation.
Let's use this formula to solve the problem.
Initial point: P(-1, 2)
Final point: P'(2, 1)
Let's substitute the given values into the formula:
2 = -1*cos(θ) - 2*sin(θ)
1 = -1*sin(θ) + 2*cos(θ)
Now we can solve this system of equations to find the angle of rotation, θ.
First, let's solve for cos(θ):
2 = -1*cos(θ) - 2*sin(θ)
2*cos(θ) = -1*cos(θ)*cos(θ) - 2*sin(θ)*cos(θ)
Next, solve for sin(θ):
1 = -1*sin(θ) + 2*cos(θ)
1*sin(θ) = -1*sin(θ)*cos(θ) + 2*cos(θ)*cos(θ)
Now, divide the two equations to find tan(θ):
(2*cos(θ))/(1*sin(θ)) = (-1*cos(θ)*cos(θ) - 2*sin(θ)*cos(θ))/(-1*sin(θ)*cos(θ) + 2*cos(θ)*cos(θ))
Simplifying the equation:
2*cos(θ)*sin(θ) = -cos(θ)*(cos(θ) + 2*sin(θ))
2*sin(θ) = -cos(θ) - 2*sin(θ)
Now, divide both sides of the equation by cos(θ) to isolate the term containing sin(θ):
2*sin(θ)/cos(θ) = -1 - 2*tan(θ)
Using the identity tan(θ) = sin(θ)/cos(θ):
2*tan(θ) = -1 - 2*tan(θ)
Rearranging the equation:
4*tan(θ) = -1
tan(θ) = -1/4
We know that tan(θ) = sin(θ)/cos(θ), so we can write:
sin(θ)/cos(θ) = -1/4
Using the trigonometric identity tan(θ) = sin(θ)/cos(θ):
tan(θ) = -1/4
Since the tangent is negative, θ must lie in the second or fourth quadrant.
Using the inverse tangent function, we can find the angle:
θ = arctan(-1/4)
Evaluating this using a calculator, we find:
θ ≈ -14.04 degrees
Therefore, the point P(-1, 2) is rotated to (2, 1) through an angle of approximately 14.04 degrees in a clockwise direction.
To describe the rotation, we need to determine the degree and direction of the rotation.
First, let's find the angle of rotation. We can do this by calculating the angle between the original point and the rotated point using trigonometry. The angle can be found using the arctangent function:
angle = arctan((y2 - y1) / (x2 - x1))
Given the original point P(x1, y1) = (-1, 2) and the rotated point P'(x2, y2) = (2, 1), the calculation becomes:
angle = arctan((1 - 2) / (2 - (-1)))
= arctan((-1) / 3)
Using a calculator or a trigonometric table, we find that arctan(-1/3) is approximately -18.43 degrees.
Now, let's determine the direction of rotation. We can determine the direction by considering the signs of the differences between the x-coordinates and y-coordinates of the original and rotated points. From the given points (-1, 2) and (2, 1), we observe that:
- The x-coordinate increased from -1 to 2 (positive change).
- The y-coordinate decreased from 2 to 1 (negative change).
This indicates that the rotation is counter-clockwise (positive direction).
In summary, the rotation is approximately -18.43 degrees counter-clockwise.