Position vector, displacement, distance

1. Position vector

WHY do we need it? "Location" is meaningless without a reference. Saying "the cat is at 3" means nothing until you say 3 metres from where, in which direction. The position vector packs both into one object.

WHAT is it? An arrow rooted at the origin, tip at the object.

HOW to compute its length? By Pythagoras in 3D: $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$

2. Displacement

Derivation from first principles. Start at $\vec{r}_i$ , end at $\vec{r}_f$ . To get from start to end we ask: "what arrow added to $\vec{r}_i$ gives $\vec{r}_f$ ?" $\vec{r}_i + \Delta\vec{r} = \vec{r}_f \;\Rightarrow\; \boxed{\Delta\vec{r} = \vec{r}_f - \vec{r}_i}$ This is just vector subtraction. Notice the origin cancels — displacement does NOT depend on where you put O. That's why displacement is "physical": two observers with different origins still agree on it.

In components: $\Delta\vec{r} = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j} + (z_f-z_i)\hat{k}$ $|\Delta\vec{r}| = \sqrt{(x_f-x_i)^2 + (y_f-y_i)^2 + (z_f-z_i)^2}$

3. Distance (path length)

WHY different from displacement? Displacement only knows endpoints; distance remembers every twist. If you walk forward 5 m then back 5 m, your displacement is $0$ but distance is $10$ m.

Figure — Position vector, displacement, distance

Common mistake Steel-manning the classic confusions

Mistake 1: "Distance and magnitude of displacement are the same." Why it feels right: On a straight one-way walk they ARE equal, and most intro examples are straight lines. The fix: The moment the path bends or reverses, distance pulls ahead. Always picture a curved/looping path to test yourself.

Mistake 2: "Displacement depends on the origin." Why it feels right: The position vector clearly changes if you move the origin, so people assume displacement does too. The fix: Displacement is a difference $\vec{r}_f - \vec{r}_i$ ; the origin appears in both terms and cancels. It is origin-independent.

Mistake 3: "Displacement can be negative, distance can be negative." Why it feels right: In 1D we write displacement like $-5$ m, so "negative distance" seems possible. The fix: The " $-$ " is a direction in 1D. Distance (a path length) is always $\ge 0$ .

Recall Feynman: explain to a 12-year-old

Imagine your home is the "start dot." A position arrow points from home to wherever you are. If you go to a friend's house and then to the park, displacement is the single straight arrow from home to the park — like a bird flying straight there. Distance is how far your feet actually walked on the roads. The bird's straight flight is always shorter than (or equal to) your walking, because roads bend. If you walk somewhere and come right back home, the bird flew nowhere (displacement zero) but your feet still got tired (distance big!).

Flashcards

What is a position vector?

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

\vec{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.

\Delta\vec r = \vec r_f - \vec r_i

(change in position vector).

Why is displacement independent of the choice of origin?

It is a difference

\vec r_f-\vec 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.

|\Delta\vec r| \le \text{distance}

When does

|\Delta\vec r| = \text{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

\ge 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?

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

Can distance be negative?

No, it is a path length, always

\ge 0

Connections

Scalars and vectors — displacement is a vector, distance is a scalar.
Vector addition and subtraction — $\Delta\vec r = \vec r_f - \vec r_i$ uses subtraction.
Average velocity and average speed — velocity uses displacement; speed uses distance.
Coordinate systems and unit vectors — $\hat i, \hat j, \hat k$ define the position vector.
Pythagoras theorem — used for magnitudes.
Motion in a straight line — the special case where distance $=|\Delta\vec r|$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, sabse pehle yeh samjho ki kisi cheez ki position batane ke liye humein ek origin (reference point) chahiye. Position vector $\vec r$ ek arrow hai jo origin se object tak jaata hai, aur usko hum $x\hat i + y\hat j + z\hat k$ likhte hain. Bina origin ke "main 3 par hoon" ka koi matlab nahi — kahan se 3, kis direction mein?

Ab displacement sirf start aur end point dekhta hai. Formula hai $\Delta\vec r = \vec r_f - \vec r_i$ — bas final minus initial. Yeh ek vector hai (magnitude + direction). Mast baat yeh hai ki displacement origin par depend nahi karta, kyunki subtraction mein origin cancel ho jaata hai. Isiliye displacement "real physical" quantity hai.

Distance alag cheez hai — yeh poora raasta (path length) measure karti hai jo tumne actually chala. Yeh ek scalar hai, hamesha positive ya zero. Agar tum aage 5 m jaake wapas 5 m aa gaye, toh displacement zero (kyunki wahi point) lekin distance 10 m (kyunki pair toh chale!).

Yaad rakhne wali sabse important baat: $|\Delta\vec r| \le \text{distance}$ . Seedhi line hamesha sabse chhoti hoti hai, isliye displacement ki length distance se kabhi zyada nahi ho sakti. Dono barabar tabhi hote hain jab motion ek hi seedhi direction mein ho, bina muде. Exam mein yeh inequality aur "round trip mein displacement zero" wale concepts bahut puchhte hain!

1. Position vector

2. Displacement

3. Distance (path length)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Measurement, Vectors & Kinematics

Connections