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# Collinear|Definition & Meaning

## Definition

Collinearity is a property of points when two or more points passing through or laying on a single or straight line. The word collinear derived from a Latin word “**col**” meaning **together** and “**linear**” meaning **same line. **The points with these property is called **collinear points.**

Collinearity in geometry is basically defined as a property of points being on a **single** and **straight line**. In a broader sense, the term has been applied to aligned objects or points in a **row**.

## Collinear Points In Maths

A collinear point is a series of **two** or **more** points along the **same linear fashion**. Collinear points are possible on multiple planes however not on multiple lines. Collinearity is the characteristic of points being collinear. So any **three** or **more** points are only collinear when they are in the relatively similar straight line. When there is just single line that can cross three different points than that points are considered **collinear**. Examine the figure below, where the collinear points are **A**, **B**, **D** and **E**.

In the above shown figure there are 4 points which are **A(2,4)**, **B(6,4)**, **D(-6,4)** and **E(-12,4)** present on a same straight line that is **y=4**. As these all points are in a row so these points are collinear points.

**Figure 2 –** Collinear points on a line y=x

Above figure is showing different points although the points are different but one thing is same for all these points is that all these points are **collinear** that is laying on the same straight line **y=x**.

### Real-life Examples

In real life, examples of collinear points include pupils standing in a line during an assembly, parked cars in a straight path in a parking lot, and a set of beads kept in a straight line. Collinear points are** useful in solving** real-world Euclidean Geometry problems, though many of them are quite complex to discuss here.

In many contexts, **collinearity simplifies things**. For example, the collinearity between two variables in **regression analysis** is a particularly helpful fact as it hints at a **large correlation** between them. Furthermore, the **collinearity of calibration points** is an important assumption in the process of **camera calibration**, where the perspective effect of cameras is guessed after a basic linear model based on collinearity.

## Formulas For Collinear Points

There are various formulas which are used to find the the collinearity between different points. But the three main formulas used to determine the collinear points are:

- Distance formula
- Area of triangle
- Slope Formula

### Distance Formula

We use the **distance formula **to calculate the distance between the **first **point** **and the **second** point, further the distance in between the **second** point and **third** points. Then we check if the **sum** of these distances are **equal** with the distance between the **first** point and the **third** points. This can only happen if the **three** points are **collinear**. The distance formula is utilized to determine the distance between two points for whom coordinates are known to us.

The distance between two points **R****(x _{1},y_{1})** and

**S**

**(x**can be determine as follow

_{2},y_{2})**d = $\mathsf{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}$**

### Area of Triangle

If** three** points are collinear, **no triangle** can be formed. We will check the triangle’s points by inserting them in the triangle area formula. If the area of a triangle is equal to zero, the points are considered collinear. In other term, the triangle made up of three collinear points has no area because it is simply a line connecting the three points. The method for the area of a triangle, which is used to check point collinearity of three points **X(x _{1},y_{1})** ,

**Y(x**and

_{2},y_{2})**Z(x**, is as follows:

_{3},y_{3})**Area = **$\mathsf{\dfrac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]}$**= 0**

### Slope Formula

We use the slope formula to calculate the slope of the lines formed by the three points under consideration. If the slopes of the three points are equal, then the three points seems to be collinear.

For instance, if we have three points **A**, **B**, and **C**, these points will be collinear if the slope of line **AB** equals the slope of line **BC **which is further equals to the slope of a line **AC**. This slope formula is considered when there is need to find the slope of a line connecting two points.

The formula of the slope joining two points is given by the following equation:

**Slope = m = (y _{2 }– y_{1}) / (x_{2 }– x_{1})**

## Example Problems on Checking Collinearity of Points

### Example 1

Find if these points **A**(−3,−1), **B**(−1,0), and **C**(1,1) are collinear.

**Solution**

These three points will be considered as collinear if the addition of distances between **AB **and **BC** is equal to distance between **CA**.

Distance between AB = ** $\mathsf{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}$**

**= $\mathsf{\sqrt{(-1+3)^2 + (0+1)^2}}$**

**= $\mathsf{\sqrt{5}}$**

Distance between BC = **$\mathsf{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}$**

**= $\mathsf{\sqrt{(1+1)^2 + (1-0)^2}}$**

**= $\mathsf{\sqrt{5}}$**

Distance between CA = **$\mathsf{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}$**

**= $\mathsf{\sqrt{(1+3)^2 + (1+1)^2}}$**

**= $\mathsf{\sqrt{20}}$**

**= 2 $\mathsf{\sqrt{5}}$**

To check collinearity,

= **$\mathsf{\sqrt{5} + \sqrt{5} = 2\sqrt{5}}$**

So point **A**,**B** and **C** are **collinear**.

**Example 2**

Find if the following points **P**(-1.5,3), **Q**(6,-2) and **R**(-3,4) of a triangle are collinear?

**Solution**

**Area of Triangle PQR = **$\mathsf{\dfrac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]}$

**= $\mathsf{\dfrac{1}{2}}$[-1.5(-2-4)+6(4-3)-3(3-(-2)]**

**= $\mathsf{\dfrac{1}{2}}$[-1.5(-6)+6(1)-3(3+2)]**

**= $\mathsf{\dfrac{1}{2}}$[-1.5(-6)+6-3(5)]**

**= $\mathsf{\dfrac{1}{2}}$[9+6-15]**

**= $\mathsf{\dfrac{1}{2}}$[0]**

**= 0**

Since the area is **zero** here so the vertices are collinear.

### Example 3

Show if the following three points **x**(1, 3),** Y**(5, 7) and **Z**(9, 11) are collinear points?

**Solution**

If these three points **X**(1, 3),** Y**(5, 7) and **Z**(9, 11) are really collinear, then the slopes of any two given points, **XY**, **YZ** & **XZ** will be equal.

Now, by using the above slope formula we can find the slopes of the given points,

Slope of XY = **(7 – 3)/ (5 – 1) = 4/4 = 1**

Slope of YZ = **(11 – 7)/ (9 – 5) = 2/2 = 1**

Slope of XZ = **(11 – 3) /(9 – 1) = 8/8 = 1**

From the above calculations, we can clearly see that the slopes of points XY, YZ, and XZ are the **same** (that is, **1)** for all given points, which means that all these three points are collinear.

*All the figures above were created with Geogebra.*