Math 2.0

Geometry Definition

Collinear points

Builds on Affine spaceAffine span

Definition

Let KK be a field and let AA be an affine space over a vector space VV over KK. A set SS of points of AA is collinear if the linear span of the vectors q−pq - p for p,q∈Sp, q \in S has dimension at most 1. Equivalently, SS is collinear if SS has at most one point or a line of AA contains SS.

Remarks

The empty set and each set with one point are collinear. Each set of two points is collinear. A set SS with at least two points is collinear if and only if its affine span is a line. In ℝ2\mathbb{R}^2, the points (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2) and (x3,y3)(x_3, y_3) are collinear if and only if (x2−x1)(y3−y1)−(x3−x1)(y2−y1)=0(x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) = 0. Mathlib's Collinear uses the first form of the definition.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Affine space, concept, no evidence levelAffine spaceconceptAffine span, concept, no evidence levelAffine spanconceptCollinear points, definition, no evidence levelCollinear pointsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Affine space concept
  2. Affine span concept
  3. Collinear points definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Collinearity, Points on a line
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WritingW1 stated
MSC51N10
DefinesCollinear points
UsesAffine space, Affine span

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