Points lying on one line are called as Collinear. Any two points are ever collinear because when connected, they determine a line.

Three or more points can be collinear but it is not necessary that all the points has to be collinear. In the above figure P, Q, and R are collinear points.

In Cartesian plane any three points are collinear (say the points are A($x_1, y_1$) B($x_2, y_2$) and C($x_3, y_3$) if the ratios

$\frac{(y_2 - y_1)}{(x_2 - x_1)}$ =

$\frac{(y_3 – y_1)}{(x_3 – x_1)}$ =

$\frac{(y_3 – y_2)}{(x_3 – x_2)}$ are equal.

In other words the slope of line formed by any two given points is same.

**Example:** Determine if the points A(2, 6), B(-1, 0) and C(5, 12) are collinear.

**Solution:**Let us determine slope formed by pair of points say A and B.

A(2, 6) B(-1, 0)

slope of line joining AB =

$\frac{(0 - 6)}{(-1 - 2)}$ =

$\frac{-6}{-3}$ Slope = 2

Check slope of line joining B and C.

B(-1, 0) and C(5, 12)

Slope of line joining B and C =

$\frac{(12 - 0)}{(5-(-1)}$ =

$\frac{12}{6}$ Slope = 2

Since slope of lines joining AB and BC are equal, the points A, B and C are collinear.

### Non-collinear points

These are the points which will not lie on the same plane, in the above figure X, Y, and Z are example for non-collinear points.

If three points are given say X Y and Z and slopes of lines formed by the pairs of points do not match then the points are not lying on the same line. Such points are called as non collinear.

**Example:** Determine if the points X(-2, -4), y(0, -1) and Z(3, 5)

**Solution:** To check if the three points are collinear let us determine the sope of lines formed by pair of points say X-Y and Y-Z.

Slope of line joining X and Y

X(-2, -4) and Y(0, -1)

slope =

$\frac{(-1 - (-4))}{(0 - (-2))}$ =

$\frac{3}{2}$ Determine the slope of line joining Y and Z

Y(0, -1) and Z(3, 5)

Slope =

$\frac{(5 - (-1))}{(3 - 0)}$ =

$\frac{6}{3}$ = 2

Since the slope of XY is not equal to the slope of YZ the points are not lying on the same line and hence are non collinear.

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