Collinear refers to the concept of three or more points that are located on the same line. In other words, when three or more points lie on a straight line, they are collinear. This is a fundamental concept in geometry and is essential in many mathematical and scientific fields. The term “collinear” comes from the Latin words “con” (together) and “linea” (line), meaning “together on a line.”
The concept of collinearity is straightforward to understand. Consider three points A, B, and C, which are not in a straight line. However, we can draw a line through points A and B, and another line through points B and C. These lines will intersect at point B. Now, if we draw a line through points A and C, we can observe that this line also intersects at point B. Therefore, points A, B, and C are collinear.
Collinearity is a crucial concept in geometry, as it enables us to determine the properties of lines, angles, and shapes. For instance, we can use collinearity to determine whether a point lies on a particular line or not. Additionally, we can use the concept of collinearity to find the equation of a line that passes through two known points.
Collinear points also play a significant role in linear algebra, where they are used to represent vectors. In linear algebra, vectors are represented as arrays of numbers, and the numbers in the arrays represent the coordinates of the vector in a given space. Vectors can be added and subtracted to obtain new vectors, and they can be multiplied by scalars to obtain new vectors with different magnitudes.
If we consider two vectors, u and v, they are collinear if they lie on the same line or are parallel to each other. Two vectors are parallel if their slopes are the same. When two vectors are collinear, one vector is a scalar multiple of the other. In other words, one vector can be obtained by multiplying the other vector by a scalar.
Collinearity is also an essential concept in linear regression, which is a statistical method used to establish a relationship between two variables. Linear regression involves fitting a line through a set of data points to determine the relationship between the independent variable (x) and the dependent variable (y).
In linear regression, the collinearity of the data points plays a crucial role in determining the accuracy of the model. If the data points are collinear, it means that they lie on the same line, and therefore, there is a strong linear relationship between the independent and dependent variables. However, if the data points are scattered, it may indicate that there is no strong relationship between the variables, and the model may not be accurate.
The concept of collinearity is also essential in computer graphics, where it is used to represent the positions of vertices and edges in 2D and 3D shapes. In computer graphics, collinearity is used to determine whether a point lies on a particular line or not. Additionally, collinearity is used to determine the intersection of lines and to calculate the area of triangles and other geometric shapes.
Collinearity is also essential in astronomy, where it is used to determine the positions of stars and other celestial bodies. Astronomers use the concept of collinearity to determine the angular distances between stars and to calculate the positions of planets and other celestial bodies in the sky.
In conclusion, collinearity is a fundamental concept in geometry, linear algebra, statistics, computer graphics, and astronomy. It is a simple yet powerful concept that enables us to determine the properties of lines, angles, and shapes. Collinearity is essential in many mathematical and scientific fields, and it is used to represent the positions of points, vectors, and
Definitions
Collinear is a term used to describe three or more points that lie on the same straight line. The term “linear” is derived from the Latin word “linea,” meaning “line.” Thus, collinear points are points that can be connected by a straight line. In simpler terms, any three or more points that can be joined by a single straight line are collinear.
Examples
Example 1: Three points on the same line In the figure below, points A, B, and C are collinear because they lie on the same straight line.
A----------------B----------------C
Example 2: Points forming a line segment In the figure below, points P, Q, and R are collinear, and they form a line segment PQ and QR.
P----------Q----------R
Example 3: Points forming a line In the figure below, points X, Y, and Z are collinear, and they form a line XYZ.
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Example 4: Three points not on the same line In the figure below, points D, E, and F are not collinear because they do not lie on the same straight line.
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Example 5: Four points on the same line In the figure below, points G, H, I, and J are collinear because they lie on the same straight line.
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Quiz
- What is collinear? a. Three or more points that lie on the same straight line b. Four or more points that lie on the same straight line c. Two or more points that lie on the same straight line
- What is the Latin word meaning line? a. Collinear b. Linear c. Linea
- Which of the following examples is not collinear? a. A, B, and C b. P, Q, and R c. D, E, and F
- In the figure below, are points K, L, and M collinear?
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a. Yes b. No
- What is the difference between collinear and coplanar? a. Collinear refers to points on the same line, and coplanar refers to points on the same plane. b. Collinear refers to points on the same plane, and coplanar refers to points on the same line. c. Collinear and coplanar mean the same thing.
- Can two points be collinear? a. Yes b. No
- In the figure below, are points N, O, P, and Q collinear?
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a. Yes b. No
- How many points are needed to determine a straight line? a. One b. Two c.
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