A Collinear Calculator helps determine if three given points are collinear. Collinear points lie on the same straight line. The calculator uses the coordinates of the three points to check if they form a straight line or not. This is particularly useful in geometry, engineering, and various fields of science.
Understanding the Calculator’s Purpose and Functionality
The Collinear Calculator checks if three points are collinear by calculating the area of the triangle formed by these points. If the area is zero, the points are collinear.
Formula:
Three points A(x1,y1)A(x_1, y_1)A(x1,y1), B(x2,y2)B(x_2, y_2)B(x2,y2), and C(x3,y3)C(x_3, y_3)C(x3,y3) are collinear if the area of the triangle formed by these points is zero. The formula to check for collinearity is:
Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\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|Area=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
For the points to be collinear:
Area=0\text{Area} = 0Area=0
Inputs:
- (x1,y1)(x_1, y_1)(x1,y1): Coordinates of point A
- (x2,y2)(x_2, y_2)(x2,y2): Coordinates of point B
- (x3,y3)(x_3, y_3)(x3,y3): Coordinates of point C
Step-by-Step Examples
Let’s check if the points A(1,2)A(1, 2)A(1,2), B(3,4)B(3, 4)B(3,4), and C(5,6)C(5, 6)C(5,6) are collinear.
- Substitute the coordinates into the formula:
Area=12∣1(4−6)+3(6−2)+5(2−4)∣\text{Area} = \frac{1}{2} \left| 1(4 – 6) + 3(6 – 2) + 5(2 – 4) \right|Area=21∣1(4−6)+3(6−2)+5(2−4)∣
- Calculate each term:
1(4−6)=1×(−2)=−21(4 – 6) = 1 \times (-2) = -21(4−6)=1×(−2)=−2
3(6−2)=3×4=123(6 – 2) = 3 \times 4 = 123(6−2)=3×4=12
5(2−4)=5×(−2)=−105(2 – 4) = 5 \times (-2) = -105(2−4)=5×(−2)=−10
- Sum the terms:
Sum=−2+12−10=0\text{Sum} = -2 + 12 – 10 = 0Sum=−2+12−10=0
- Calculate the absolute value and multiply by 12\frac{1}{2}21:
Area=12∣0∣=0\text{Area} = \frac{1}{2} \left| 0 \right| = 0Area=21∣0∣=0
Since the area is zero, the points A(1,2)A(1, 2)A(1,2), B(3,4)B(3, 4)B(3,4), and C(5,6)C(5, 6)C(5,6) are collinear.
Relevant Information Table
| Point | Coordinates |
|---|---|
| A | (1, 2) |
| B | (3, 4) |
| C | (5, 6) |
Conclusion: Benefits and Applications of the Calculator
The Collinear Calculator is a useful tool for quickly checking if three points lie on the same line. This has practical applications in various fields:
- Geometry: Helps in solving problems related to lines and planes.
- Engineering: Useful in designing structures and ensuring components align correctly.
- Geography: Helps in mapping and determining straight-line distances between points.
- Computer Graphics: Ensures alignment of graphical elements.
By automating the calculation process, the Collinear Calculator saves time and reduces the likelihood of errors, making it an invaluable tool for professionals and students alike.