A point is rotated about the origin. Its coordinates are P(−4,10) and P′(−10,−4). Determine the degree of rotation and direction by applying rotation mapping.
To determine the degree of rotation and direction, we can use the formula for finding the angle between two vectors in the Cartesian plane.
The vector from the origin to the point P is given by OP = (-4, 10).
The vector from the origin to the point P' is given by OP' = (-10, -4).
To find the angle between these two vectors, we can use the dot product formula:
OP · OP' = |OP| |OP'| cos(theta)
The dot product of two vectors can be found by multiplying their corresponding components and summing the results:
(-4)(-10) + (10)(-4) = sqrt((-4)^2 + (10)^2) sqrt((-10)^2 + (-4)^2) cos(theta)
40 - 40 = sqrt(116) sqrt(116) cos(theta)
0 = 116 cos(theta)
Since the dot product is zero, we can conclude that the vectors are perpendicular, and therefore, the angle between them is 90 degrees.
Since the vectors are pointing in opposite directions, we can determine the direction of rotation by considering which half-plane (left or right) the rotation occurs in. In this case, the rotation is from the positive y-axis to the negative x-axis, so the direction of rotation is clockwise.
Therefore, the degree of rotation is 90 degrees in the clockwise direction.
To determine the degree of rotation and direction, we can use the formula for rotation mapping. Rotation mapping is represented by the equation:
x' = x*cos(theta) - y*sin(theta)
y' = x*sin(theta) + y*cos(theta)
In this case, the coordinates of the original point P are (-4, 10) and its rotated coordinates P' are (-10, -4).
Comparing the rotation mapping equation to the given coordinates, we can write:
-10 = -4*cos(theta) - 10*sin(theta)
-4 = -4*sin(theta) + 10*cos(theta)
To solve these equations, we can divide the second equation by 2:
-2 = -sin(theta) + 5*cos(theta)
Now, let's square both equations and add them:
(-10)^2 + (-4)^2 = (-4*cos(theta) - 10*sin(theta))^2 + (-4*sin(theta) + 10*cos(theta))^2
Simplifying this equation, we get:
116 = 116*cos^2(theta) + 116*sin^2(theta)
Since cos^2(theta) + sin^2(theta) = 1, we can simplify further:
116 = 116
This shows that the equations are satisfied for any value of theta, meaning there are infinitely many possible rotations that can result in the given coordinates. Therefore, we cannot determine the specific degree of rotation and direction.