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What is the Collinearity of Three Points using Vectors?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
Collinearity of three points means that these three points lie on the same straight line. Using vectors, we can check if three points A, B, and C are collinear by seeing if vector AB is a scalar multiple of vector BC (or AC, or any other pair formed by these points).
Simple Example
Quick Example
Imagine three friends, Rohan, Priya, and Sameer, standing in a straight line for a school assembly. If Rohan is at one end, Sameer is in the middle, and Priya is at the other end, they are collinear. If Sameer steps out of the line, they are no longer collinear.
Worked Example
Step-by-Step
Let's check if points A(1, 2), B(3, 6), and C(5, 10) are collinear using vectors.
Step 1: Find the vector AB. To do this, subtract the coordinates of A from B. Vector AB = (3-1, 6-2) = (2, 4).
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Step 2: Find the vector BC. To do this, subtract the coordinates of B from C. Vector BC = (5-3, 10-6) = (2, 4).
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Step 3: Check if vector AB is a scalar multiple of vector BC. Here, vector AB = (2, 4) and vector BC = (2, 4). We can see that (2, 4) = 1 * (2, 4).
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Step 4: Since vector AB is 1 times vector BC (a scalar multiple), and they share a common point B, the points A, B, and C are collinear.
Answer: Yes, the points A(1, 2), B(3, 6), and C(5, 10) are collinear.
Why It Matters
Understanding collinearity is crucial in fields like AI/ML for pattern recognition and image processing, helping computers 'see' straight lines. In Engineering, it's used to design stable structures and ensure parts align correctly. Even in Space Technology, calculating satellite trajectories often involves checking if certain points are collinear for precise navigation.
Common Mistakes
MISTAKE: Students calculate two vectors (e.g., AB and BC) and find one is a scalar multiple of the other, but forget to check for a common point. | CORRECTION: Always ensure the two vectors you compare share a common point (like B in vector AB and vector BC) for the three points to be collinear.
MISTAKE: Incorrectly subtracting coordinates to find the vector components, leading to wrong vector values. | CORRECTION: Remember to subtract the initial point's coordinates from the terminal point's coordinates (e.g., for vector PQ, it's Q_x - P_x and Q_y - P_y).
MISTAKE: Assuming that if two vectors are parallel, the points are automatically collinear without considering their relative positions. | CORRECTION: While parallel vectors are a necessary condition, collinearity requires them to be parallel AND share a common point. If they don't share a point, they are just parallel lines.
Practice Questions
Try It Yourself
QUESTION: Are the points P(1, 1), Q(2, 3), and R(3, 5) collinear? | ANSWER: Yes
QUESTION: Given points A(2, 5), B(4, 9), and C(6, 12). Are they collinear? | ANSWER: No
QUESTION: If points D(1, -2), E(3, y), and F(5, 10) are collinear, find the value of y. | ANSWER: y = 4
MCQ
Quick Quiz
Which condition proves that points P, Q, and R are collinear?
Vector PQ is perpendicular to Vector QR
Vector PQ is equal to Vector PR
Vector PQ is a scalar multiple of Vector QR
The sum of the lengths of PQ and QR is greater than PR
The Correct Answer Is:
C
For points P, Q, and R to be collinear, the vector formed by two of them (like PQ) must be parallel to the vector formed by another pair (like QR), and they must share a common point. Being a scalar multiple indicates parallelism and shared direction/line.
Real World Connection
In the Real World
Think about how your GPS app on your mobile phone works. When you're navigating from your home to a new chai shop, the app often calculates the shortest, straightest route. This involves checking if key turning points or landmarks are 'collinear' with the path, ensuring the route is a single, continuous line segment. Delivery apps like Zepto also use this to optimize delivery paths, making sure their riders follow a straight line between stops when possible.
Key Vocabulary
Key Terms
VECTOR: A quantity having both magnitude and direction, like a path from one point to another. | SCALAR: A quantity that only has magnitude, like a number (e.g., 2, -5, 0.5). | MAGNITUDE: The length or size of a vector. | ORIGIN: The starting point (0,0) in a coordinate system. | COORDINATES: A set of numbers that show an exact position on a map or graph.
What's Next
What to Learn Next
Great job understanding collinearity! Next, you should explore 'Section Formula using Vectors'. This concept will teach you how to find the coordinates of a point that divides a line segment in a specific ratio, which builds directly on your vector knowledge and is super useful for geometry problems.
