Answer :
If the vectors \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form four right angles, we need to determine the correct relationship between these vectors. Here is a thorough step-by-step explanation:
1. Understanding Four Right Angles Intersecting:
- When two lines or vectors intersect to form four right angles (90 degrees each), it implies that each pair of angles formed around the intersection point is orthogonal.
- In simpler terms, if \(\overrightarrow{P Q}\) intersects \(\overrightarrow{R S}\) and creates right angles, then \(\overrightarrow{P Q}\) must be perpendicular to \(\overrightarrow{R S}\).
2. Analyzing the Given Statements:
- Option A: \(\overrightarrow{Q Q}\) and \(\overrightarrow{R S}\) are skew.
- This is incorrect terminology and doesn't make sense in this context. "\(\overrightarrow{Q Q}\)" seems to be a typographical error or not relevant for our vectors.
- Option B: \(\overrightarrow{P Q} \perp \overleftrightarrow{R S}\)
- This means that \(\overrightarrow{P Q}\) is perpendicular to \(\overrightarrow{R S}\). Given our understanding that the vectors intersect to form right angles, this statement is correct.
- Option C: \(\overrightarrow{P Q} = \overrightarrow{R S}\)
- This states that the vectors are equal, which cannot be concluded just from the intersection forming right angles. They only need to be perpendicular, not equal.
- Option D: \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are parallel.
- This statement is incorrect because if they were parallel, they could not intersect to form four right angles.
3. Conclusion:
- The correct statement derived from the given situation is that \(\overrightarrow{P Q}\) is perpendicular to \(\overrightarrow{R S}\).
Thus, the correct choice is:
B. [tex]\(\overrightarrow{P Q} \perp \overleftrightarrow{R S}\)[/tex]
1. Understanding Four Right Angles Intersecting:
- When two lines or vectors intersect to form four right angles (90 degrees each), it implies that each pair of angles formed around the intersection point is orthogonal.
- In simpler terms, if \(\overrightarrow{P Q}\) intersects \(\overrightarrow{R S}\) and creates right angles, then \(\overrightarrow{P Q}\) must be perpendicular to \(\overrightarrow{R S}\).
2. Analyzing the Given Statements:
- Option A: \(\overrightarrow{Q Q}\) and \(\overrightarrow{R S}\) are skew.
- This is incorrect terminology and doesn't make sense in this context. "\(\overrightarrow{Q Q}\)" seems to be a typographical error or not relevant for our vectors.
- Option B: \(\overrightarrow{P Q} \perp \overleftrightarrow{R S}\)
- This means that \(\overrightarrow{P Q}\) is perpendicular to \(\overrightarrow{R S}\). Given our understanding that the vectors intersect to form right angles, this statement is correct.
- Option C: \(\overrightarrow{P Q} = \overrightarrow{R S}\)
- This states that the vectors are equal, which cannot be concluded just from the intersection forming right angles. They only need to be perpendicular, not equal.
- Option D: \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are parallel.
- This statement is incorrect because if they were parallel, they could not intersect to form four right angles.
3. Conclusion:
- The correct statement derived from the given situation is that \(\overrightarrow{P Q}\) is perpendicular to \(\overrightarrow{R S}\).
Thus, the correct choice is:
B. [tex]\(\overrightarrow{P Q} \perp \overleftrightarrow{R S}\)[/tex]
