Straight Lines in 3D

Definition: Straight Line

A straight line in 3D space is an 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 Angle 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

$$\angle (a;b)$$

Definition: Perpendicularity

Two straight lines in 3D are perpendicular if the angle between them is a right angle.

Properties

Theorem: Intersecting Lines $⟹$ 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 parallel to $a$.

$$\forall P,\exists !b:a\parallel b\wedge P\in b$$

PROOF

TODO

Theorem: Parallel Lines $⟹$ Plane

If two straight lines in 3D space are 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 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.

$$a\parallel b\wedge b\parallel c⟹a\parallel c$$

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.

$$\angle (a_1,b_1)=\angle (a_2,b_2)$$

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 $a^{\prime }$ and $b^{\prime }$ such that $a$ and $a^{\prime }$ are parallel and $b$ and $b^{\prime }$ are also parallel.