What is the Collinearity Test for Three Points? | Simple Explanation for Class 12

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What is the Collinearity Test for Three Points using Vectors?

Grade Level:

Class 12

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Definition

What is it?

The collinearity test for three points using vectors helps us check if three given points lie on the same straight line. If three points A, B, and C are collinear, it means the vector AB will be parallel to vector BC, and they will share a common point B.

Simple Example

Quick Example

Imagine three friends, Rohan, Priya, and Amit, standing in a line for a bus. If Rohan is at point A, Priya at point B, and Amit at point C, they are collinear. Using vectors, if the 'direction' from Rohan to Priya is the same as the 'direction' from Priya to Amit, they are all in a straight line.

Worked Example

Step-by-Step

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

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

---Step 2: Find the vector BC.
Vector BC = Position vector of C - Position vector of B
BC = (3 - 2)i + (4 - 3)j + (5 - 4)k = 1i + 1j + 1k

---Step 3: Check if AB is a scalar multiple of BC.
We need to see if AB = λ * BC, where λ is a scalar (a simple number).
Here, (1i + 1j + 1k) = 1 * (1i + 1j + 1k).
So, λ = 1.

---Step 4: Conclude.
Since vector AB is a scalar multiple of vector BC (AB = 1 * BC), and they share a common point B, the points A, B, and C are collinear.

Answer: Points A, B, and C are collinear.

Why It Matters

Understanding collinearity is crucial in fields like AI/ML for path planning of self-driving cars, ensuring they stay on a straight road. In engineering, it helps design structures where forces act along a straight line. Future engineers and data scientists use this concept to make things work precisely.

Common Mistakes

MISTAKE: Only checking if vectors are parallel without ensuring they share a common point. | CORRECTION: For collinearity, two vectors formed by the three points (e.g., AB and BC) must be parallel AND share a common point (B in this case).

MISTAKE: Calculating AB and AC, and finding they are parallel, but not checking the scalar multiple correctly. | CORRECTION: If you use AB and AC, then AC must be a scalar multiple of AB (AC = λ AB). Also, A must be the common point. Make sure the scalar multiple applies to all components (i, j, k).

MISTAKE: Confusing collinearity with coplanarity. | CORRECTION: Collinearity means points are on the same straight LINE. Coplanarity means points are on the same FLAT SURFACE (plane). These are different concepts.

Practice Questions

Try It Yourself

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

QUESTION: Determine if the points A(1, -2, 3), B(2, -3, 4), and C(3, -4, 6) are collinear. | ANSWER: No, they are not collinear.

QUESTION: For what value of 'x' are the points D(1, 2, 3), E(3, 4, 5), and F(x, 6, 7) collinear? | ANSWER: x = 5

MCQ

Quick Quiz

If vector PQ = 2i + 3j - k and vector QR = 4i + 6j - 2k, what can we say about points P, Q, and R?

They are not collinear.

They are collinear.

They form a right-angled triangle.

They are coplanar but not collinear.

The Correct Answer Is:

Vector QR is 2 times vector PQ (4i + 6j - 2k = 2 * (2i + 3j - k)). Since QR is a scalar multiple of PQ and they share a common point Q, the points P, Q, R are collinear.

Real World Connection

In the Real World

In building construction, engineers use collinearity tests to ensure that the support pillars are perfectly aligned in a straight line, preventing structural weaknesses. Similarly, when laying railway tracks or designing roads in India, surveyors use these principles to ensure straight paths, which is vital for safety and smooth travel.

Key Vocabulary

Key Terms

VECTOR: A quantity having both magnitude and direction, like displacement or velocity. | SCALAR MULTIPLE: When one vector is obtained by multiplying another vector by a number (scalar). | COLLINEAR: Points that lie on the same straight line. | POSITION VECTOR: A vector that represents the position of a point in space relative to an origin.

What's Next

What to Learn Next

Next, you can explore the concept of coplanarity, which is about points lying on the same plane (a flat surface). This builds on understanding vectors and will help you visualize 3D space even better, useful for advanced geometry and physics.