Position vector, displacement, distance — Hinglish notes

1.1.13 · Physics › Measurement, Vectors & Kinematics

1. Position vector

WHY isko zaroort hai? "Location" bina reference ke meaningless hai. "Cat 3 pe hai" kehna kuch nahi batata jab tak yeh na bolo ki 3 metre kahan se, kis direction mein. Position vector dono ko ek cheez mein pack kar deta hai.

WHAT hai yeh? Ek arrow jo origin pe rooted hai, tip object pe hai.

HOW karte hain iska length compute? 3D mein Pythagoras se: $∣ r ∣ = x^{2} + y^{2} + z^{2}$

2. Displacement

First principles se derivation. $r_{i}$ pe start karo, $r_{f}$ pe end karo. Start se end tak pahunchne ke liye hum poochte hain: "kaunsa arrow $r_{i}$ mein add hoke $r_{f}$ deta hai?" $r_{i} + Δ r = r_{f} \Rightarrow Δ r = r_{f} - r_{i}$ Yeh sirf vector subtraction hai. Dhyaan do ki origin cancel ho jaata hai — displacement is baat pe depend nahi karta ki tumne O kahan rakha. Isliye displacement "physical" hai: do observers jinke alag-alag origins hain woh phir bhi isme agree karte hain.

Components mein: $Δ r = (x_{f} - x_{i}) \hat{i} + (y_{f} - y_{i}) \hat{j} + (z_{f} - z_{i}) \hat{k}$ $∣Δ r ∣ = (x_{f} - x_{i})^{2} + (y_{f} - y_{i})^{2} + (z_{f} - z_{i})^{2}$

3. Distance (path length)

WHY displacement se alag hai? Displacement sirf endpoints jaanta hai; distance har modhh yaad rakhta hai. Agar tum 5 m aage chalo phir 5 m wapas, tumhara displacement $0$ hai lekin distance $10$ m hai.

Figure — Position vector, displacement, distance

Common mistake Classic confusions ko steel-man karna

Galti 1: "Distance aur displacement ka magnitude same hote hain." Kyun sahi lagta hai: Seedhe one-way walk mein woh HOTE HAIN equal, aur zyaadatar intro examples seedhi lines hain. Fix: Jis pal path bend ya reverse kare, distance aage nikal jaati hai. Khud ko test karne ke liye hamesha curved/looping path socho.

Galti 2: "Displacement origin pe depend karta hai." Kyun sahi lagta hai: Position vector clearly change ho jaata hai agar origin move karo, toh log assume karte hain displacement bhi karta hai. Fix: Displacement ek difference $r_{f} - r_{i}$ hai; origin dono terms mein aata hai aur cancel ho jaata hai. Yeh origin-independent hai.

Galti 3: "Displacement negative ho sakta hai, distance negative ho sakti hai." Kyun sahi lagta hai: 1D mein hum displacement $- 5$ m likhte hain, toh "negative distance" possible lagti hai. Fix: " $-$ " 1D mein ek direction hai. Distance (ek path length) hamesha $\geq 0$ hoti hai.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tumhara ghar "start dot" hai. Ek position arrow ghar se wahan point karta hai jahan tum ho. Agar tum dost ke ghar gaye aur phir park gaye, displacement ghar se park tak ka single seedha arrow hai — jaise ek chidiya seedha wahan udi ho. Distance woh hai kitna tumhare pair actually roads pe chale. Chidiya ki seedhi flight hamesha tumhare chalne se chhoти hoti hai (ya barabar), kyunki roads bend karti hain. Agar tum kahan gaye aur wapas ghar aa gaye, chidiya kahin nahi udi (displacement zero) lekin tumhare pair phir bhi thak gaye (distance badi!).

Flashcards

What is a position vector?

The directed arrow from the origin O to a point P;

r = x \hat{i} + y \hat{j} + z \hat{k}

Why does location require a reference point?

A coordinate is meaningless without an origin and axes to measure direction & magnitude from.

Define displacement as a formula.

Δ r = r_{f} - r_{i}

(change in position vector).

Why is displacement independent of the choice of origin?

It is a difference

r_{f} - r_{i}

; the origin appears in both vectors and cancels.

Define distance.

The total length of the actual path travelled; a scalar that never decreases.

State the inequality linking distance and displacement.

∣Δ r ∣ \leq distance

When does

∣Δ r ∣ = distance

Only for straight-line motion in a single direction (no turning or reversing).

Is displacement a scalar or vector?

Vector (has magnitude AND direction).

Is distance a scalar or vector?

Scalar (magnitude only, always

\geq 0

For a closed loop (start = end), what is the displacement?

Zero vector, even though distance is positive.

Magnitude of displacement from A(1,2) to B(4,6) in metres?

3^{2} + 4^{2} = 5

Can distance be negative?

No, it is a path length, always

\geq 0

Connections

Scalars and vectors — displacement ek vector hai, distance ek scalar.
Vector addition and subtraction — $Δ r = r_{f} - r_{i}$ subtraction use karta hai.
Average velocity and average speed — velocity displacement use karta hai; speed distance use karta hai.
Coordinate systems and unit vectors — $\hat{i}, \hat{j}, \hat{k}$ position vector define karte hain.
Pythagoras theorem — magnitudes ke liye use hota hai.
Motion in a straight line — woh special case jahan distance $= ∣Δ r ∣$ .

1. Position vector

2. Displacement

3. Distance (path length)

Flashcards

Connections

Concept Map