What is Displacement Vector?

S7-SA2-0063

Grade Level:

Class 12

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

Definition

What is it?

A displacement vector is a vector that shows the shortest straight-line distance and direction from an object's initial position to its final position. It tells you 'how far and in what direction' an object has moved, regardless of the path taken.

Simple Example

Quick Example

Imagine you walk from your home to the neighbourhood chai shop, then to a friend's house, and finally back home. Your total path covered is long. However, your displacement vector for the entire journey would be zero, because your final position (home) is the same as your initial position (home).

Worked Example

Step-by-Step

Let's say a delivery rider starts their journey from point A (their shop) which is at coordinates (2, 3) on a map. --- They deliver a parcel and end up at point B (customer's house) which is at coordinates (8, 7). --- We want to find the displacement vector from A to B. --- The displacement vector (let's call it 'd') is found by subtracting the initial position vector from the final position vector. --- Initial position vector (r_A) = (2i + 3j). Final position vector (r_B) = (8i + 7j). --- Displacement vector d = r_B - r_A = (8i + 7j) - (2i + 3j). --- d = (8-2)i + (7-3)j. --- So, the displacement vector d = 6i + 4j. This means the rider moved 6 units in the x-direction and 4 units in the y-direction.

Why It Matters

Understanding displacement vectors is crucial for designing self-driving cars in EVs and for guiding rockets in Space Technology. Engineers use it to calculate precise movements, and even in Medicine, doctors track the displacement of internal organs during surgery. It helps in predicting where things will end up.

Common Mistakes

MISTAKE: Confusing displacement with distance. | CORRECTION: Displacement is the shortest straight line from start to end, with direction. Distance is the total path covered, without direction.

MISTAKE: Forgetting that displacement is a vector quantity. | CORRECTION: Always remember to include the direction (e.g., using 'i' and 'j' components, or 'North', 'East') along with the magnitude.

MISTAKE: Incorrectly subtracting coordinates (e.g., final - initial). | CORRECTION: Always subtract the initial position vector from the final position vector: Displacement = Final Position - Initial Position.

Practice Questions

Try It Yourself

QUESTION: A student walks from their classroom (position (1, 2)) to the library (position (5, 2)). What is their displacement vector? | ANSWER: 4i + 0j (or just 4i)

QUESTION: An auto-rickshaw starts at the origin (0, 0), drives 3 km East, then 4 km North. What is its final displacement vector from the origin? | ANSWER: 3i + 4j

QUESTION: A drone flies from point P (10, 5, 2) to point Q (15, 8, 4). Calculate the displacement vector of the drone. | ANSWER: 5i + 3j + 2k

MCQ

Quick Quiz

Which of the following describes a displacement vector?

Total path length covered by an object

The shortest straight-line distance and direction from start to end

The speed of an object in a particular direction

The rate of change of an object's position

The Correct Answer Is:

A displacement vector specifically refers to the shortest straight path from an initial to a final point, including direction. Options A, C, and D describe distance, velocity, and speed respectively, which are different concepts.

Real World Connection

In the Real World

When you order food using an app like Zomato or Swiggy, the app's navigation system calculates the displacement vector from the restaurant to your house. This helps drivers find the most direct route, even if they take a slightly longer path to avoid traffic. This is also how ISRO scientists track satellites!

Key Vocabulary

Key Terms

VECTOR: A quantity having both magnitude and direction | POSITION VECTOR: A vector from the origin to a point in space | MAGNITUDE: The size or amount of a vector, without direction | INITIAL POSITION: The starting point of an object | FINAL POSITION: The ending point of an object

What's Next

What to Learn Next

Now that you understand displacement, you should learn about 'Velocity Vector'. Velocity is simply how fast an object's displacement changes over time, adding another layer to understanding motion.