What is Position Vector?

S7-SA2-0062

Grade Level:

Class 12

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

Definition

What is it?

A position vector is a vector that tells us the exact location of a point in space relative to a fixed starting point, called the origin. Think of it like a GPS for a single point, always starting from 'home base' (the origin).

Simple Example

Quick Example

Imagine your school gate is the 'origin' (0,0). If your friend's classroom is 50 meters east and 30 meters north from the gate, the vector pointing from the gate directly to their classroom is the position vector of their classroom. It tells you exactly where their classroom is starting from the gate.

Worked Example

Step-by-Step

PROBLEM: Find the position vector of a point P (3, 4) from the origin O (0, 0).

STEP 1: Identify the coordinates of the point and the origin.
Point P = (3, 4)
Origin O = (0, 0)

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STEP 2: Recall that a position vector of a point (x, y) from the origin is given by x * i-hat + y * j-hat (where i-hat and j-hat are unit vectors along the x and y axes).

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STEP 3: Substitute the coordinates of point P into the formula.
Position vector of P = 3 * i-hat + 4 * j-hat

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STEP 4: This vector represents the displacement from the origin to point P.

ANSWER: The position vector of point P (3, 4) is 3i + 4j.

Why It Matters

Position vectors are super important for understanding movement and location in various fields. Engineers use them to design self-driving cars and drones, ensuring they know exactly where they are. In AI/ML, they help robots navigate and understand their environment, making them useful in factories and even for delivering food, creating exciting career opportunities.

Common Mistakes

MISTAKE: Confusing position vector with displacement vector. | CORRECTION: A position vector always starts from the origin (0,0) and points to a specific point. A displacement vector shows the change in position between ANY two points, not necessarily from the origin.

MISTAKE: Writing a position vector as just (x, y) coordinates without i-hat and j-hat. | CORRECTION: While (x, y) are coordinates, a position vector is typically expressed using unit vectors, like xi + yj, to clearly indicate it's a vector quantity with direction.

MISTAKE: Assuming the origin is always (0,0) without checking. | CORRECTION: For most problems, the origin is (0,0) by default. However, always confirm if a different reference point is specified as the origin. If it is, calculate the vector from that new origin.

Practice Questions

Try It Yourself

QUESTION: What is the position vector of a point A (5, -2) from the origin? | ANSWER: 5i - 2j

QUESTION: If the position vector of point B is -3i + 7j, what are the coordinates of point B? | ANSWER: (-3, 7)

QUESTION: A drone takes off from a launchpad (origin). After some time, its position is given by the vector 100i + 50j (in meters). If it then moves 20 meters west and 30 meters south, what is its new position vector? | ANSWER: (100 - 20)i + (50 - 30)j = 80i + 20j

MCQ

Quick Quiz

Which of the following best describes a position vector?

A vector showing the distance between two points.

A vector that starts from the origin and points to a specific location.

A scalar quantity indicating location.

A vector showing only the direction of a point.

The Correct Answer Is:

A position vector specifically defines the location of a point with respect to a fixed origin, giving both magnitude (distance from origin) and direction. Option A describes a displacement vector. Option C is wrong as vectors are not scalars. Option D is incomplete as it lacks magnitude.

Real World Connection

In the Real World

When you order food using an app like Zomato or Swiggy, the app uses position vectors! Your phone's GPS determines your current position vector from a global origin. The delivery rider's location is also tracked using position vectors. This helps the app calculate the shortest route and estimated delivery time, making sure your hot samosas reach you quickly!

Key Vocabulary

Key Terms

VECTOR: A quantity having both magnitude and direction, like velocity or force. | ORIGIN: The fixed starting point (usually (0,0)) from which position vectors are measured. | COORDINATES: A set of values (like x, y) that show the exact position of a point. | UNIT VECTOR: A vector with a magnitude of 1, used to show direction (like i-hat for x-axis, j-hat for y-axis).

What's Next

What to Learn Next

Great job understanding position vectors! Next, you should explore 'Displacement Vectors' and 'Vector Addition'. These build directly on position vectors and will help you understand how things move from one position to another, which is crucial for physics and engineering.