To determine the degree of rotation and direction for point \( P(-4, 10) \) when it transforms to point \( P'(-10, -4) \), we can use the concept of rotating a point in the coordinate plane.
-
Understanding Rotation: The formula for rotating a point \( (x, y) \) around the origin by an angle \( \theta \) (in degrees) is given by: \[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \]
-
Setting Up the Points:
- Original Point \( P(-4, 10) \) corresponds to \( (x, y) = (-4, 10) \).
- Transformed Point \( P'(-10, -4) \) corresponds to \( (x', y') = (-10, -4) \).
-
Calculating the Angle of Rotation: We can find the angle \( \theta \) by analyzing how the coordinates change. To determine this, we can find the lengths of the vectors and their relative angles.
- Calculate the original and final lengths using the distance formula:
- Original Length \( r = \sqrt{(-4)^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116} \)
- Final Length \( r' = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \)
The lengths are equal, confirming that \( P \) was rotated instead of translated.
- Calculate the original and final lengths using the distance formula:
-
Finding the Angle Between the Vectors: We can use the dot product to find the cosine of the angle between the original and new coordinates.
\[ \text{dot product} = (-4)(-10) + (10)(-4) = 40 - 40 = 0 \]
Since the dot product is zero, the angle between the two vectors is \( 90^\circ \).
-
Determining Direction: To find the direction of rotation, we can observe how the point \( P(-4, 10) \) moves to \( P'(-10, -4) \):
- The original point is in the second quadrant and the new point is in the third quadrant, which indicates a counter-clockwise rotation.
Conclusion
The point \( P(-4, 10) \) is rotated by \( 90^\circ \) in the counter-clockwise direction to arrive at the point \( P'(-10, -4) \).
8 answers