Statistics Definitions > Collinear Definition

## Definition

Collinear points (also called colinear) are points which lie on one straight line.

In the above image, points A, F, and B are collinear, as they lie on a straight line. G and H are not collinear to A, F, and B.

Two points are always collinear, as one can always draw a straight line through them. Three points may or may not be colinear; if they are colinear, the triangle which is formed with the three points as vertices will have an area of zero.

## Demonstrating Collinearity of Points

In an ideal world, one could demonstrate and determine collinearity by drawing a line through points on a graph. In the real world, though, this is not quite enough: every point is infinitely small, so a line may appear to go through a dot on the graph which doens’t actually go through the point.

Another, better way to determine colinear points in 2D is to use that property we just learned: the area of the triangle they make is zero. We know how to calculate the area of a triangle when we know the vertices:

We can take out the ½ and rearrange to make a simpler equation:

if our points satisfy this equation, they are colinear. Alternatively, if the area of the triangle turns out to be anything other than zero, we’ve proven that our points are not colinear.

As an example of this, consider the three points (5,4), (6,2), and (3,4). Plugging this into the area formula, we have

[5(2 – 4)+ 6(4 – 4) + 3(4 – 2)]/2 = [-10 + 6]/2 = 4/2 = 2

Of course, if we’re given a linear equation that all of our points satisfy, we also know that they are colinear. For instance, if we are given the line y = 3 x + 2, we know that the points (2,8), (1,5), and (0,2) are all collinear.

## References

Wilson, J. EMAT 4690/6690 Class Notes: Lesson 1. Retrieved from http://jwilson.coe.uga.edu/emat6680fa08/eckstein/instructionalunit/day1/day1.html on Feb 20, 2019

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