What is Collinearity using Vectors? | Simple Explanation for Class 12

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What is the Condition for Collinearity using Vectors?

Grade Level:

Class 12

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Definition

What is it?

For three points to be collinear using vectors, it means they all lie on the same straight line. The main condition is that the vector connecting the first two points must be a scalar multiple of the vector connecting the last two points (or any two pairs of points). This scalar multiple can be any real number, positive or negative.

Simple Example

Quick Example

Imagine three chai stalls, A, B, and C, along a straight road. If the vector from stall A to B is exactly twice the vector from stall B to C, but in the same direction, then all three stalls are on the same straight road. This means they are collinear.

Worked Example

Step-by-Step

Let's check if points A(1, 2, 3), B(3, 4, 7), and C(7, 8, 15) are collinear using vectors.

Step 1: Find vector AB.
Vector AB = Position vector of B - Position vector of A
AB = (3i + 4j + 7k) - (1i + 2j + 3k)
AB = (3-1)i + (4-2)j + (7-3)k
AB = 2i + 2j + 4k

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Step 2: Find vector BC.
Vector BC = Position vector of C - Position vector of B
BC = (7i + 8j + 15k) - (3i + 4j + 7k)
BC = (7-3)i + (8-4)j + (15-7)k
BC = 4i + 4j + 8k

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Step 3: Check if AB is a scalar multiple of BC (or vice versa).
We see that BC = 4i + 4j + 8k. If we factor out 2, we get BC = 2(2i + 2j + 4k).

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Step 4: Compare with AB.
Since AB = 2i + 2j + 4k, we can write BC = 2 * AB.

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Answer: Yes, since vector BC is a scalar multiple (2) of vector AB, points A, B, and C are collinear.

Why It Matters

Understanding collinearity is crucial in computer graphics for rendering straight lines and in robotics for path planning to ensure movements are efficient. Engineers use it to design structures, while AI systems in self-driving cars use it to predict if other vehicles are moving in a straight line relative to them, ensuring safety on Indian roads.

Common Mistakes

MISTAKE: Calculating only one vector (e.g., AB) and not comparing it with another vector (e.g., BC or AC). | CORRECTION: You need at least two vectors formed by the three points to check their relationship.

MISTAKE: Assuming vectors are collinear if they are parallel, without checking if they share a common point. | CORRECTION: For points to be collinear, the vectors formed between them must not only be parallel (scalar multiples) but also share a common point (e.g., B in AB and BC).

MISTAKE: Forgetting that the scalar multiple (lambda) can be negative. | CORRECTION: A negative scalar just means the vectors point in opposite directions, but they are still parallel and can make points collinear if they share a common point.

Practice Questions

Try It Yourself

QUESTION: Are points P(1,1), Q(2,3), and R(3,5) collinear? | ANSWER: Yes

QUESTION: Given points A(2, -1, 3), B(4, 2, 1), and C(8, 8, -3). Are they collinear? | ANSWER: Yes

QUESTION: If points A(x, 1), B(3, 2), and C(5, 3) are collinear, find the value of x. | ANSWER: x = 1

MCQ

Quick Quiz

For three points A, B, and C to be collinear, which condition must be true?

Vector AB is perpendicular to vector BC

Vector AB is equal to vector BC

Vector AB is a scalar multiple of vector BC

Vector AB + Vector BC = 0

The Correct Answer Is:

For points to be collinear, the vectors formed by them must be parallel and share a common point. Being a scalar multiple means they are parallel. Option B is too restrictive, as they don't have to be equal in magnitude.

Real World Connection

In the Real World

When a drone delivers a package for a service like Swiggy or Zomato, its navigation system uses vector collinearity. If it needs to fly in a straight line from the warehouse (A) to a checkpoint (B) and then to the customer's home (C), the system ensures that the path from A to B is a scalar multiple of the path from B to C. This ensures a straight, efficient delivery route.

Key Vocabulary

Key Terms

COLLINEAR: Points lying on the same straight line | VECTOR: A quantity having both magnitude and direction | SCALAR MULTIPLE: A vector multiplied by a real number, changing its magnitude and possibly direction | POSITION VECTOR: A vector from the origin to a point

What's Next

What to Learn Next

Next, you can explore the concept of coplanarity, which is about points lying on the same plane, similar to how collinearity deals with points on the same line. This will help you understand 3D space even better.