Skew Lines in Geometry

Definition of Skew Lines

• Skew lines are lines that are not parallel and do not intersect.
• They must also be non-coplanar, meaning they do not lie in the same plane.

Characteristics of Skew Lines

• Skew lines can be easily identified in diagrams of three-dimensional figures.
• They can be edges of a geometric figure or free-floating lines in space.

Conditions for Skew Lines

• There are three conditions for skew lines:
  • They are not parallel.
  • They do not intersect.
  • They are not coplanar.

Representation of Lines in Three Dimensions

• Lines in three dimensions can be written in vector, parametric, or symmetric form.

Configurations of Skew Lines

• A configuration of skew lines is a set of lines in which all lines are skew.
• A configuration can have many lines that are all skewed to each other.

Distance Between Skew Lines

• The distance between two skew lines is the shortest distance between them, which is equal to the length of the perpendicular between them.
• The shortest distance can be calculated using the cross-product of the vectors describing the two lines.

Gallucci's Theorem

• Gallucci's Theorem states that if three skew lines all meet three other skew lines, then any transversal of the first three will meet any transversal of the other three.

Testing for Skewness

• Two lines can be tested for skewness by showing that they are not parallel and not intersecting.
• Parallel lines have the same "slope" and intersecting lines can be checked by solving a system of equations.

Skew lines are non-parallel and non-intersecting lines that are not coplanar, often seen in 3D figures. Learn about skew lines and how to identify them in diagrams.