Question
Planes Q and R are parallel. Explain how you know lines a and b are skew.
Planes Q and R are parallel. Line a is on plane Q and is diagonal down and to the left. Line b is on plane R and is diagonal up and to the left.
Planes Q and R are parallel. Line a is on plane Q and is diagonal down and to the left. Line b is on plane R and is diagonal up and to the left.
Answers
GPT-4o mini
To determine if lines a and b are skew, we need to recall the properties of skew lines. Skew lines are lines that do not intersect and are not parallel; that is, they exist in different planes and do not share any points.
Given that planes Q and R are parallel, here are the reasons we can ascertain that lines a and b are skew:
1. **Different Planes**: Line a is located on plane Q, while line b is located on plane R. Since these planes are distinct and parallel, lines a and b cannot lie in the same plane. Therefore, they meet the first criterion for being skew: they are in different planes.
2. **Non-Intersecting**: Since lines a and b lie on parallel planes Q and R, and one line is moving diagonally down and to the left (line a) while the other is moving diagonally up and to the left (line b), they will never intersect. If they were to intersect, they would have to be in the same plane, which contradicts the fact that they are on different parallel planes.
3. **Not Parallel**: Lines a and b are also not parallel to each other. Even though they move diagonally in a similar direction (up and to the left for b and down and to the left for a), they lie in different planes, which means they can’t be parallel lines as defined in three-dimensional geometry.
In summary, lines a and b are skew because they are located on parallel planes (meaning they cannot intersect), they do not coincide, and their directional paths do not allow for any intersection point while remaining on their respective planes.
Given that planes Q and R are parallel, here are the reasons we can ascertain that lines a and b are skew:
1. **Different Planes**: Line a is located on plane Q, while line b is located on plane R. Since these planes are distinct and parallel, lines a and b cannot lie in the same plane. Therefore, they meet the first criterion for being skew: they are in different planes.
2. **Non-Intersecting**: Since lines a and b lie on parallel planes Q and R, and one line is moving diagonally down and to the left (line a) while the other is moving diagonally up and to the left (line b), they will never intersect. If they were to intersect, they would have to be in the same plane, which contradicts the fact that they are on different parallel planes.
3. **Not Parallel**: Lines a and b are also not parallel to each other. Even though they move diagonally in a similar direction (up and to the left for b and down and to the left for a), they lie in different planes, which means they can’t be parallel lines as defined in three-dimensional geometry.
In summary, lines a and b are skew because they are located on parallel planes (meaning they cannot intersect), they do not coincide, and their directional paths do not allow for any intersection point while remaining on their respective planes.