Straight Lines in 3D#
Definition: Straight Line
A straight line in 3D space is an [[TODO|unbounded]] locus of points which is homeomorphic to the real numbers \(\mathbb{R}\).
Definition: Angle between Straight Lines
Let \(a\) and \(b\) be straight lines in 3D.
The angle between \(a\) and \(b\) is the Inner Product Spaces between any pair of vectors such that one of the vectors is between points which lie on \(a\) and the other is between points which lie on \(b\).
Notation
Definition: Perpendicularity
Two straight lines in 3D are perpendicular if the angle between them is a right angle.
Properties#
Theorem: Intersecting Lines \(\implies\) Plane
For each pair of intersecting straight lines there exists a unique plane which contains them.
Proof
TODO
Parallel Lines#
Theorem: Parallel Lines and Points
If \(a\) is a straight line in 3D space, then for each point \(P\) there exists a unique straight line \(b\) which goes through \(P\) and is [[TODO|parallel]] to \(a\).
Proof
TODO
Theorem: Parallel Lines \(\implies\) Plane
If two straight lines in 3D space are [[TODO|parallel]], then there exists a unique plane which contains both of them.
Proof
TODO
Theorem: Parallel Lines and Plane Intersections
If two straight lines in 3D space are [[TODO|parallel]], then every plane which intersects one of them also intersects the other.
Proof
TODO
Theorem: Three Parallel Lines
Let \(a,b,c\) be straight lines in 3D space.
If \(a\) and \(b\) are parallel and \(b\) and \(c\) are parallel, then \(a\) and \(c\) are also parallel.
Proof
TODO
Skew Lines#
Definition: Skew Lines
Skew lines are two straight lines in 3D space which do not lie in the same plane.
Theorem: Angle between Skew Lines
Let \(a\) and \(b\) be skew lines, let \(a_1\) and \(a_2\) be parallel to \(a\) and let \(b_1\) and \(b_2\) be parallel to \(b\).
If \(a_1\) and \(b_1\) intersect and \(a_2\) and \(b_2\) also intersect, then the angles they form are equal.
Proof
TODO
Definition: Angle between Skew Lines
The angle between two skew lines \(a\) and \(b\) is the angle between any pair of intersecting lines \(a'\) and \(b'\) such that \(a\) and \(a'\) are parallel and \(b\) and \(b'\) are also parallel.