Collinearity of three points
Collinearity of Three Points
Collinearity in geometry refers to the property of points lying on a single straight line. When we talk about the collinearity of three points, we are essentially asking whether these points share a common line on which they all lie. This concept is important in both two-dimensional and three-dimensional geometry, as well as in vector analysis.
Understanding Collinearity
To determine if three points are collinear, we can use various methods depending on the information given. These methods include:
- Slope Method: In a two-dimensional plane, if the slope between each pair of points is the same, then the points are collinear.
- Area of Triangle Method: If the area of the triangle formed by three points is zero, then the points are collinear.
- Vector Method: In two or three dimensions, if the vectors formed by two pairs of points are proportional, the points are collinear.
Slope Method
The slope of a line segment between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a two-dimensional plane is given by:
$$ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} $$
For three points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ to be collinear, the slope of AB must equal the slope of BC, which must also equal the slope of AC.
Area of Triangle Method
The area of a triangle with vertices at points $A$, $B$, and $C$ can be calculated using the determinant formula:
$$ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| $$
If the area is zero, then the points are collinear.
Vector Method
Given three points $A$, $B$, and $C$, we can form two vectors $\vec{AB}$ and $\vec{AC}$. If $\vec{AB}$ is a scalar multiple of $\vec{AC}$, then the points are collinear. In other words, there exists a scalar $k$ such that:
$$ \vec{AB} = k \vec{AC} $$
Table of Differences and Important Points
| Criteria | Slope Method | Area of Triangle Method | Vector Method |
|---|---|---|---|
| Dimension | 2D only | 2D only | 2D or 3D |
| Formula | $\frac{y_2 - y_1}{x_2 - x_1}$ | $\frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |
| Applicability | Only when x-coordinates are not equal | Always applicable in 2D | Always applicable in 2D and 3D |
| Limitations | Undefined for vertical lines | None | None |
Examples
Example 1: Slope Method
Consider three points $A(1, 2)$, $B(3, 6)$, and $C(5, 10)$. To check for collinearity using the slope method:
- Slope of AB: $\frac{6 - 2}{3 - 1} = 2$
- Slope of BC: $\frac{10 - 6}{5 - 3} = 2$
- Slope of AC: $\frac{10 - 2}{5 - 1} = 2$
Since all slopes are equal, points A, B, and C are collinear.
Example 2: Area of Triangle Method
Using the same points $A(1, 2)$, $B(3, 6)$, and $C(5, 10)$, the area of the triangle can be calculated as:
$$ \text{Area} = \frac{1}{2} \left| 1(6 - 10) + 3(10 - 2) + 5(2 - 6) \right| = 0 $$
Since the area is zero, the points are collinear.
Example 3: Vector Method
For points $A(1, 2, 3)$, $B(2, 4, 6)$, and $C(3, 6, 9)$ in three-dimensional space:
- Vector $\vec{AB} = \langle 2 - 1, 4 - 2, 6 - 3 \rangle = \langle 1, 2, 3 \rangle$
- Vector $\vec{AC} = \langle 3 - 1, 6 - 2, 9 - 3 \rangle = \langle 2, 4, 6 \rangle$
Since $\vec{AB} = \frac{1}{2} \vec{AC}$, the points A, B, and C are collinear.
In conclusion, understanding the concept of collinearity and the methods to determine it is crucial for solving problems in geometry and vector analysis. The choice of method may depend on the given information and the context of the problem.