Answer :
To determine the relationship between the two vectors \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) when they intersect to form four right angles, we need to understand what these geometrical terms imply.
Here are the possible situations and the correct logical deduction step-by-step:
1. Right Angles and Perpendicularity:
- When two lines (or vectors) intersect at a right angle, it means they form a 90-degree angle at the point of intersection.
- If they form four right angles at the intersection, each of the four angles between the vectors is 90 degrees.
- This implies that each of the angles between \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) and its continuation (extended indefinitely in both directions) forms perpendicular lines.
2. Option Analysis:
- Option A: \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are parallel:
- Parallel vectors never intersect. If \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) were parallel, they wouldn't form any angles between each other.
- Thus, this option is incorrect.
- Option B: \(\overleftrightarrow{P Q} \perp \overleftrightarrow{R S}\):
- This states that the lines are perpendicular. Perpendicular lines intersect to form right angles.
- Since the problem states that \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form four right angles, this would mean they are indeed perpendicular.
- Thus, this option is correct.
- Option C: \(\overleftrightarrow{P Q}\) and \(\overparen{R S}\) are skew:
- Skew lines are lines that do not lie in the same plane and thus never intersect.
- If \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect, they cannot be skew.
- Thus, this option is incorrect.
- Option D: \(\overleftrightarrow{P Q} = \overleftrightarrow{R S}\):
- This states that the lines are identical, meaning every point on \(\overleftrightarrow{P Q}\) lies on \(\overleftrightarrow{R S}\) and vice versa.
- This is not necessarily true if they are only intersecting at one point to form right angles.
- Thus, this option is incorrect.
Given the analysis above, the correct statement is:
B. [tex]\(\overleftrightarrow{P Q} \perp \overleftrightarrow{R S}\)[/tex]
Here are the possible situations and the correct logical deduction step-by-step:
1. Right Angles and Perpendicularity:
- When two lines (or vectors) intersect at a right angle, it means they form a 90-degree angle at the point of intersection.
- If they form four right angles at the intersection, each of the four angles between the vectors is 90 degrees.
- This implies that each of the angles between \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) and its continuation (extended indefinitely in both directions) forms perpendicular lines.
2. Option Analysis:
- Option A: \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) are parallel:
- Parallel vectors never intersect. If \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) were parallel, they wouldn't form any angles between each other.
- Thus, this option is incorrect.
- Option B: \(\overleftrightarrow{P Q} \perp \overleftrightarrow{R S}\):
- This states that the lines are perpendicular. Perpendicular lines intersect to form right angles.
- Since the problem states that \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form four right angles, this would mean they are indeed perpendicular.
- Thus, this option is correct.
- Option C: \(\overleftrightarrow{P Q}\) and \(\overparen{R S}\) are skew:
- Skew lines are lines that do not lie in the same plane and thus never intersect.
- If \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect, they cannot be skew.
- Thus, this option is incorrect.
- Option D: \(\overleftrightarrow{P Q} = \overleftrightarrow{R S}\):
- This states that the lines are identical, meaning every point on \(\overleftrightarrow{P Q}\) lies on \(\overleftrightarrow{R S}\) and vice versa.
- This is not necessarily true if they are only intersecting at one point to form right angles.
- Thus, this option is incorrect.
Given the analysis above, the correct statement is:
B. [tex]\(\overleftrightarrow{P Q} \perp \overleftrightarrow{R S}\)[/tex]
