Parallel Lines and Planes in Space

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What defines parallel lines in a three-dimensional space?
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Parallel lines in a three-dimensional space are lines that are in the same plane and do not intersect.
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What defines parallel planes in space?
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Parallel planes are two planes in space that do not intersect, regardless of how far they are extended.
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How can you determine if two lines are parallel using vectors?
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Two lines are parallel if their direction vectors are scalar multiples of each other.
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What is a line that is parallel to a plane?
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A line is parallel to a plane if it never intersects the plane.
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How can you identify when a line is parallel to a plane using the normal vector?
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A line is parallel to a plane if its direction vector is perpendicular to the normal vector of the plane.
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Can two planes be parallel if they have different normal vectors? Why or why not?
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No, two planes cannot be parallel if they have different normal vectors, as parallel planes must have proportional normal vectors.
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What role do angles play in determining parallelism in space?
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Two lines or planes are parallel if the angle between them (or their direction vectors) is zero.
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Can skew lines be parallel? Why?
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No, skew lines cannot be parallel because they are not in the same plane and do not intersect.
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How would you describe skew lines?
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Skew lines are lines that do not intersect and are not in the same plane.
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How can a plane in space be defined using points?
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A plane can be defined by three non-collinear points or one point and a normal vector.
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What is the relationship between the perpendicular distance and parallel planes?
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Parallel planes have a constant perpendicular distance from one another.
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How can the equation of a plane help determine parallelism?
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If two planes have normal vectors that are scalar multiples, their equations suggest that they are parallel.
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Which mathematical tool is often used to determine parallelism in 3D space?
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Vectors and their properties are used to determine parallelism in 3D space.
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What is the significance of the cross product for parallelism?
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If the cross product of two vectors is zero, the vectors are parallel.
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How can transformations affect parallel lines or planes?
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Transformations like translation preserve parallelism, while others like rotation or reflection might not maintain it.
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