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.

Answer 1

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.