Sure, let's break down the solution step-by-step.
1. Translation Vector:
- A translation involves moving every point of a figure or a space by the same distance in a specified direction.
- Here, the translation vector is [tex]\((10, -4)\)[/tex]. This means we move every point 10 units to the right (since the x-component is positive) and 4 units down (since the y-component is negative).
2. Rotation:
- A rotation is a circular movement of an object around a center (or point) of rotation.
- A non-rigid transformation means that the shape or size of the object changes when it is rotated.
- According to the given information, this specific rotation is non-rigid. This means that after the rotation, the shape and size of the object would have changed.
3. Independence of Events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex]:
- Two events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] are independent if the occurrence of one does not affect the probability of the other occurring.
- According to the information, [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] are independent events.
4. Mutually Exclusive Events:
- Two events are mutually exclusive if they cannot both occur at the same time.
- From the information, [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] are mutually exclusive events. This implies that if [tex]\(E_1\)[/tex] occurs, [tex]\(E_2\)[/tex] cannot occur, and vice versa. Hence, the intersection of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] is empty.
5. Complementary Event:
- A complementary event consists of all outcomes that are not in the event.
- According to the information provided, the complementary event contains only one element, indicating that the event and its complement together represent the entire sample space.
- This means if [tex]\(E\)[/tex] is an event, [tex]\(E^c\)[/tex] (the complement of [tex]\(E\)[/tex]) should have just one outcome not in [tex]\(E\)[/tex].
So, summing all this up in the context of each part given:
- The translation vector [tex]\((10, -4)\)[/tex] represents a translation of 10 units to the right and 4 units down.
- The rotation described is a non-rigid transformation, meaning the object changes shape and size after the rotation.
- Events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] are independent, meaning the occurrence of one doesn't affect the occurrence of the other.
- Events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] are mutually exclusive, meaning they cannot both happen at the same time.
- The complementary event contains only one element, indicating simplicity in the context of event and its complement comprising the entire sample space.
This completes the detailed step-by-step breakdown of the solution.
