Parallel Lines and Planes in Space

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What does it mean for two lines in space to be parallel?
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Two lines are parallel if they lie in the same plane and do not intersect, or if they are coplanar and equidistant from each other.
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What is a parallel plane?
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Two planes are parallel if they do not intersect and are equidistant at all points.
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How can you determine if a line is parallel to a plane?
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A line is parallel to a plane if they do not intersect or if the line is contained in a plane parallel to the given plane.
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What is the symbol used to denote parallelism between lines or planes?
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The symbol is || (e.g., line a || line b, or plane α || plane β).
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How do you prove that two lines are parallel in space?
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To prove two lines are parallel, show they have the same direction vector or that the angles between them and a transversal line are equal.
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Can two skew lines be parallel to the same plane?
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No, skew lines do not lie in the same plane, so they cannot both be parallel to the same plane.
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What is the condition for two planes to be perpendicular?
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Two planes are perpendicular if the angle between them is 90 degrees, or if a normal vector to one plane is parallel to the other plane.
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What role do vectors play in determining parallelism in space?
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Vectors help describe the direction of lines or planes; if direction vectors are scalar multiples, they indicate parallelism.
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How does the concept of parallelism extend to three dimensions?
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In 3D, parallelism involves coplanarity and equidistance; lines must not intersect, and planes must maintain constant distance.
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Can a line be parallel to two different planes?
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Yes, a line can be parallel to two different planes if it lies in a plane that is parallel to both.
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What is a real-life example of parallel planes?
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Floors in a multi-story building are real-life examples of parallel planes.
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How can parallel planes be represented in equations?
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Parallel planes have equations of the form ax + by + cz = d1 and ax + by + cz = d2, where the coefficients of x, y, and z are the same.
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What happens if two parallel planes intersect a third plane?
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The lines of intersection will be parallel to each other.
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What is the significance of parallelism in space geometry?
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Parallelism simplifies calculations and construction, aiding in understanding shapes and relationships in space.
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What tools are often used to prove parallelism in space?
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Tools include vector algebra, coordinate geometry, and geometric postulates or theorems.
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