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:

1. 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.

2. 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. Cross-Product Method

The cross-product method is another way to find collinear points. It involves calculating the cross-product of the vectors formed by each pair of points and checking if the cross-products are zero. If the cross-products are zero, then the points are collinear.

Steps:

1. Calculate the cross-product of the vectors formed by each pair of points using the formula:

$$cross-product = (x2 - x1) * (y2 - y1) - (y2 - y1) * (x2 - x1)$$

where (x1, y1) and (x2, y2) are the coordinates of the two points.

2. Check if the cross-products are zero. If the cross-products 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 cross-product of the vectors formed by each pair of points:

cross-product1 = (3 - 1) * (4 - 2) - (4 - 2) * (3 - 1) = 0 cross-product2 = (5 - 3) * (6 - 4) - (6 - 4) * (5 - 3) = 0 cross-product3 = (7 - 5) * (8 - 6) - (8 - 6) * (7 - 5) = 0

Since all the cross-products 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:

1. Create a matrix A by arranging the coordinates of the points as rows:

   $$A = [[x1, y1, 1], [x2, y2, 1], [x3, y3, 1], …]$$

2. Calculate the determinant of the matrix A.

3. 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 y-intercept.

Non-Collinear Points

• Non-collinear points are points that do not lie on the same straight line.
• They are also known as non-colinear points.
• The term “non-collinear” comes from the Latin words “non” (meaning “not”) and “linearis” (meaning “line”).
• Non-collinear points cannot be represented by a single equation of the form y = mx + b.

Examples of Collinear and Non-Collinear Points

• Collinear points:
  • The points (1, 2), (3, 4), and (5, 6) are collinear because they lie on the line y = x + 1.
• Non-collinear points:
  • The points (1, 2), (3, 4), and (5, 7) are non-collinear because they do not lie on the same straight line.

Applications of Collinear and Non-Collinear Points

• Collinear points are used in a variety of applications, such as:
  • Surveying
  • Navigation
  • Engineering
  • Architecture
• Non-collinear points are also used in a variety of applications, such as:
  • Computer graphics
  • Robotics
  • Virtual reality
  • Augmented reality

Collinear and non-collinear 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

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.