# Collinearity

Collinearity is a mathematical term used in geometry and linear algebra .

In geometry, points that lie on a straight line are called collinear. The collinearity of points played in both the affine geometry as well as in projective geometry an important role since it is invariant under certain when collineations designated pictures is.

## Collinear vectors

In linear algebra collinearity means in vectors of a vector space , that the plane defined by these vectors subspace which dimension has first If only two vectors different from the zero vector are considered, collinearity means that - to put it simply - each of the two vectors is multiplied by a scalar , i. H. a (directionless) number , into which the other vector can be converted and both vectors are linearly dependent according to the following equation : ${\ displaystyle \ beta \ in \ mathbb {R}}$ ${\ displaystyle {\ vec {a}} = \ beta \ cdot {\ vec {b}} \ Leftrightarrow {\ vec {a}} - \ beta \ cdot {\ vec {b}} = {\ vec {0} }}$ If you let the two vectors begin at the origin of the coordinates, they both lie on a straight line, so both point in the same (or exactly opposite) direction and generally have different lengths.

Collinearity investigations are often carried out when investigating the positional relationships between several straight lines . Straight lines with collinear direction vectors are either identical or “genuinely” parallel .