What is Collinearity of Three Points using Determinants?

S7-SA2-0043

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Collinearity of three points means that these three points lie on the same straight line. We can check if three points are collinear using determinants by calculating the 'area' formed by these points. If the determinant value is zero, the points are collinear.

Simple Example

Quick Example

Imagine three friends standing in a line for a selfie. If their positions can be represented by points, and they are perfectly in a straight line, then they are collinear. If one friend steps slightly forward or backward, they are no longer collinear, and the 'area' they form would be a tiny triangle, not zero.

Worked Example

Step-by-Step

Let's check if points A(1, 2), B(3, 4), and C(5, 6) are collinear using determinants.

Step 1: Write down the coordinates in the determinant form. For points (x1, y1), (x2, y2), and (x3, y3), the determinant is:
| x1 y1 1 |
| x2 y2 1 |
| x3 y3 1 |

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Step 2: Substitute the given coordinates into the determinant:
| 1 2 1 |
| 3 4 1 |
| 5 6 1 |

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Step 3: Calculate the determinant. We'll expand along the first row:
= 1 * (4*1 - 6*1) - 2 * (3*1 - 5*1) + 1 * (3*6 - 5*4)

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Step 4: Perform the multiplications and subtractions:
= 1 * (4 - 6) - 2 * (3 - 5) + 1 * (18 - 20)

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Step 5: Simplify the terms:
= 1 * (-2) - 2 * (-2) + 1 * (-2)

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Step 6: Calculate the final sum:
= -2 + 4 - 2
= 0

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Answer: Since the determinant value is 0, the points A(1, 2), B(3, 4), and C(5, 6) are collinear.

Why It Matters

Understanding collinearity is crucial in fields like AI/ML for data analysis and computer graphics, where points need to be aligned for efficient processing. Engineers use it to design structures or plan routes, ensuring components are properly aligned. It helps in developing self-driving cars to detect lane markers and ensure safe navigation.

Common Mistakes

MISTAKE: Forgetting the third column of '1's in the determinant matrix. | CORRECTION: Always remember that for a 3x3 determinant to check collinearity, the third column must consist of ones.

MISTAKE: Making calculation errors when expanding the determinant (e.g., sign errors). | CORRECTION: Double-check each step of the determinant expansion, especially the signs (plus-minus-plus) for each term.

MISTAKE: Concluding points are collinear if the determinant is a very small non-zero number (e.g., 0.0001). | CORRECTION: For perfect collinearity, the determinant must be exactly zero. Any non-zero value, no matter how small, means the points form a triangle (however tiny).

Practice Questions

Try It Yourself

QUESTION: Are points P(2, 3), Q(4, 5), and R(6, 7) collinear? | ANSWER: Yes

QUESTION: Find the value of 'k' if the points A(1, 0), B(k, 2), and C(3, 4) are collinear. | ANSWER: k = 2

QUESTION: Three points are given as (a, b+c), (b, c+a), and (c, a+b). Show that these points are collinear. | ANSWER: The determinant will be 0, proving collinearity.

MCQ

Quick Quiz

If three points are collinear, what will be the value of the determinant formed by their coordinates?

-1

Any non-zero number

The Correct Answer Is:

When three points are collinear, they do not form a triangle. Since the absolute value of the determinant gives twice the area of the triangle formed by the points, a zero determinant means zero area, implying the points lie on a single line.

Real World Connection

In the Real World

Imagine a drone delivering a package in a city like Bengaluru. For the drone to fly in a straight path from the warehouse to your home, the navigation system uses concepts like collinearity. It checks if the current position, the next waypoint, and the destination are all 'collinear' to ensure a direct, efficient flight path, much like how food delivery apps like Zomato or Swiggy plan routes.

Key Vocabulary

Key Terms

COLLINEAR: Lying on the same straight line | DETERMINANT: A special number calculated from a square matrix | COORDINATES: A set of values that show an exact position on a graph | MATRIX: A rectangular array of numbers, symbols, or expressions

What's Next

What to Learn Next

Great job understanding collinearity! Next, you can explore how determinants are used to find the area of a triangle, which is a direct application of what you just learned. This will further strengthen your understanding of geometric concepts and their algebraic representation.