Maths Collinear Points
Collinear Points
Collinear points are points that lie on the same straight line. In geometry, three or more points are said to be collinear if they lie on the same line.
Properties of Collinear Points
 The distance between any two collinear points is the sum of the distances between each point and the point in between them.
 The midpoint of any two collinear points is also collinear with them.
 If three points are collinear, then the area of the triangle formed by them is zero.
 If four points are collinear, then the sum of the opposite angles is 180 degrees.
Applications of Collinear Points
Collinear points are used in a variety of applications, including:
 Surveying: Collinear points are used to establish straight lines for surveying purposes.
 Construction: Collinear points are used to align objects during construction.
 Engineering: Collinear points are used to design and build structures.
 Computer graphics: Collinear points are used to create lines and other shapes in computer graphics.
Examples of Collinear Points
There are many examples of collinear points in the real world. Some examples include:
 The vertices of a triangle are collinear.
 The points on a line segment are collinear.
 The points on a ray are collinear.
 The points on a plane are collinear if they lie on the same line.
Collinear points are a fundamental concept in geometry. They have a variety of properties and applications in the real world.
Methods to find Collinear Points
Collinear points are points that lie on the same straight line. There are several methods to find collinear points in a set of points. Here are some common methods:
1. Slope Method
The slope method is a simple and efficient way to find collinear points. It involves calculating the slope of the line passing through each pair of points and checking if the slopes are equal. If the slopes are equal, then the points are collinear.
Steps:

Calculate the slope of the line passing through each pair of points using the formula:
$$slope = (y2  y1) / (x2  x1)$$
where (x1, y1) and (x2, y2) are the coordinates of the two points.

Check if the slopes are equal. If the slopes are equal, then the points are collinear.
Example:
Consider the following set of points:
(1, 2), (3, 4), (5, 6), (7, 8)
To find collinear points, we calculate the slope of the line passing through each pair of points:
slope1 = (4  2) / (3  1) = 1 slope2 = (6  4) / (5  3) = 1 slope3 = (8  6) / (7  5) = 1
Since all the slopes are equal, the points are collinear.
2. CrossProduct Method
The crossproduct method is another way to find collinear points. It involves calculating the crossproduct of the vectors formed by each pair of points and checking if the crossproducts are zero. If the crossproducts are zero, then the points are collinear.
Steps:
 Calculate the crossproduct of the vectors formed by each pair of points using the formula:
$$crossproduct = (x2  x1) * (y2  y1)  (y2  y1) * (x2  x1)$$
where (x1, y1) and (x2, y2) are the coordinates of the two points.
 Check if the crossproducts are zero. If the crossproducts are zero, then the points are collinear.
Example:
Consider the following set of points:
(1, 2), (3, 4), (5, 6), (7, 8)
To find collinear points, we calculate the crossproduct of the vectors formed by each pair of points:
crossproduct1 = (3  1) * (4  2)  (4  2) * (3  1) = 0 crossproduct2 = (5  3) * (6  4)  (6  4) * (5  3) = 0 crossproduct3 = (7  5) * (8  6)  (8  6) * (7  5) = 0
Since all the crossproducts are zero, the points are collinear.
3. Determinant Method
The determinant method is a more general method to find collinear points. It involves calculating the determinant of the matrix formed by the coordinates of the points. If the determinant is zero, then the points are collinear.
Steps:

Create a matrix A by arranging the coordinates of the points as rows:
$$A = [[x1, y1, 1], [x2, y2, 1], [x3, y3, 1], …]$$

Calculate the determinant of the matrix A.

Check if the determinant is zero. If the determinant is zero, then the points are collinear.
Example:
Consider the following set of points:
(1, 2), (3, 4), (5, 6), (7, 8)
To find collinear points, we create the matrix A:
$$ A = \begin{bmatrix} 1 & 2 & 1 \\ 3 & 4 & 1 \\ 5 & 6 & 1 \\ 7 & 8 & 1 \end{bmatrix} $$
The determinant of the matrix A is:
$det(A) = (1 * (4 * 1  6 * 1)  2 * (3 * 1  5 * 1) + 1 * (3 * 6  4 * 5)) = 0$
Since the determinant is zero, the points are collinear.
Conclusion
These are some common methods to find collinear points in a set of points. The choice of method depends on the specific problem and the available information.
Difference between Collinear and Non Collinear Point
Collinear Points
 Collinear points are points that lie on the same straight line.
 They are also known as colinear points.
 The term “collinear” comes from the Latin words “com” (meaning “with”) and “linearis” (meaning “line”).
 Collinear points can be represented by a single equation of the form y = mx + b, where m is the slope of the line and b is the yintercept.
NonCollinear Points
 Noncollinear points are points that do not lie on the same straight line.
 They are also known as noncolinear points.
 The term “noncollinear” comes from the Latin words “non” (meaning “not”) and “linearis” (meaning “line”).
 Noncollinear points cannot be represented by a single equation of the form y = mx + b.
Examples of Collinear and NonCollinear Points
 Collinear points:
 The points (1, 2), (3, 4), and (5, 6) are collinear because they lie on the line y = x + 1.
 Noncollinear points:
 The points (1, 2), (3, 4), and (5, 7) are noncollinear because they do not lie on the same straight line.
Applications of Collinear and NonCollinear Points
 Collinear points are used in a variety of applications, such as:
 Surveying
 Navigation
 Engineering
 Architecture
 Noncollinear points are also used in a variety of applications, such as:
 Computer graphics
 Robotics
 Virtual reality
 Augmented reality
Collinear and noncollinear points are two important concepts in geometry. They have a variety of applications in different fields, such as surveying, navigation, engineering, architecture, computer graphics, robotics, virtual reality, and augmented reality.
Collinear points vs coplanar points
Collinear Points
 Definition: Collinear points are points that lie on the same straight line.
 Properties:
 Collinear points can be extended indefinitely in both directions.
 The distance between any two collinear points is the sum of the distances between the points and any other point on the line.
 The angle between any two collinear points is 0 degrees.
 Examples:
 The points (1, 2), (3, 4), and (5, 6) are collinear.
 The points (0, 0), (1, 1), and (2, 2) are collinear.
 The points (1, 0), (0, 1), and (1, 1) are not collinear.
Coplanar Points
 Definition: Coplanar points are points that lie in the same plane.
 Properties:
 Coplanar points can be extended indefinitely in any direction within the plane.
 The distance between any two coplanar points is the shortest distance between the points that lies in the plane.
 The angle between any two coplanar points is the angle formed by the lines connecting the points to any other point in the plane.
 Examples:
 The points (1, 2, 3), (4, 5, 6), and (7, 8, 9) are coplanar.
 The points (0, 0, 0), (1, 1, 1), and (2, 2, 2) are coplanar.
 The points (1, 0, 0), (0, 1, 0), and (0, 0, 1) are not coplanar.
Comparison of Collinear and Coplanar Points
Property  Collinear Points  Coplanar Points 

Definition  Points that lie on the same straight line  Points that lie in the same plane 
Properties  Can be extended indefinitely in both directions  Can be extended indefinitely in any direction within the plane 
Distance between points  Sum of the distances between the points and any other point on the line  Shortest distance between the points that lies in the plane 
Angle between points  0 degrees  Angle formed by the lines connecting the points to any other point in the plane 
Examples  (1, 2), (3, 4), (5, 6)  (1, 2, 3), (4, 5, 6), (7, 8, 9) 
Collinear Points FAQs
What are collinear points?
Collinear points are points that lie on the same straight line. In other words, they are points that are in a straight line.
How do you determine if points are collinear?
There are a few ways to determine if points are collinear. One way is to use the slope formula. If the slope of the line between two points is the same as the slope of the line between two other points, then the four points are collinear.
Another way to determine if points are collinear is to use the distance formula. If the distance between two points is the same as the distance between two other points, then the four points are collinear.
What are some examples of collinear points?
Some examples of collinear points include:
 The points (0, 0), (1, 1), and (2, 2) are collinear because they lie on the line y = x.
 The points (0, 0), (0, 1), and (0, 2) are collinear because they lie on the line x = 0.
 The points (1, 1), (2, 2), and (3, 3) are collinear because they lie on the line y = x + 1.
What are some applications of collinear points?
Collinear points have a number of applications in geometry and other fields. Some examples include:
 Collinear points can be used to determine the slope of a line.
 Collinear points can be used to find the equation of a line.
 Collinear points can be used to determine the distance between two points.
 Collinear points can be used to find the area of a triangle.
Conclusion
Collinear points are a fundamental concept in geometry. They have a number of applications in geometry and other fields.