1.1.6
Intuition 30-second pictureKuch physical quantities ko describe karne ke liye bas ek number aur ek unit kaafi hoti hai — "5 kg", "20 °C", "3 seconds". Doosron ko woh number aur ek direction bhi chahiye hoti hai tab baat banti hai — "5 km north jao", "10 N ki force daayein taraf lagao".
Poora fark sirf itna hai: kya is quantity ke liye direction matter karti hai? Agar haan → vector . Agar nahi → scalar .
Intuition Motivating problemSochiye main 3 m chalta hoon, phir 4 m chalta hoon. Main shuru se kitni door hoon?
Agar dono walks ek hi line mein hain, toh answer 7
Agar doosri walk pehli ke perpendicular hai, toh answer 5
Agar main wapas apni taraf hi chal pada, toh answer 1
Same do numbers (3 aur 4), teen alag answers — kyunki direction ne result badal diya . Is tarah ki quantity ko ordinary numbers ki tarah add nahi kar sakte. Physics ko ek aisa label chahiye jo bataye "dhyan raho, direction matter karti hai". Woh label hai vector . Wo quantities jahan 3 + 4 hamesha 7 scalars hain.
Scalar ek aisi physical quantity hai jo poori tarah se sirf magnitude (ek number) aur ek unit se specify ho jaati hai. Yeh ordinary arithmetic (+ , − , × , ÷ Examples: mass, time, temperature, speed, distance, energy, work, charge, density, electric potential.
Vector ek aisi physical quantity hai jise magnitude aur direction dono chahiye, aur jo triangle / parallelogram law ke according add hoti hai (simple arithmetic se nahi).
Examples: displacement, velocity, acceleration, force, momentum, electric field, weight.
Common mistake "Jis cheez mein direction ho woh vector hai" — Steel-manYeh sahi kyun lagta hai: direction vectors ki headline feature hai, isliye yeh kaafi lagti hai.
Yeh incomplete kyun hai: electric current mein direction hoti hai (wire ke along) aur magnitude bhi hoti hai (amps), phir bhi yeh ek scalar hai! Junction par currents simple numbers ki tarah add hoti hain (I 1 + I 2 = I 3 Fix: asli test addition rule hai. Ek sachcha vector triangle law of addition follow karta hai. Sirf direction kaafi nahi hoti.
Property
Scalar
Vector
Specified by
magnitude + unit
magnitude + unit + direction
Adds by
ordinary arithmetic
triangle/parallelogram law
Negative meaningful hota hai?
sign = "zero se kam"
sign = opposite direction
Notation
m , t , T v , F v
Magnitude likhne ka tarika
value khud
$
Worked example Example 1 — Distance vs DisplacementAap 3 km East chalte ho, phir 4 km North.
Distance (scalar) = 3 + 4 = 7 Yeh step kyun? Distance sirf path length ka total hai; direction ignore hoti hai, isliye plain addition kaam karti hai.Displacement (vector) magnitude = 3 2 + 4 2 = 5 Yeh step kyun? Dono legs perpendicular hain (θ = 9 0 ∘ cos θ = 0 A 2 + B 2 Takeaway: same trip, scalar 7 deta hai, vector 5 deta hai. Yeh gap hi proof hai ki direction matter karti hai.
Worked example Example 3 — "Tricky" current case (Steel-man in action)Teen wires ek junction par milti hain: 2 A aur 3 A andar aati hain, bahar jaane wali current nikaliye.
Answer: 2 + 3 = 5 plain addition , bhaley hi har current apni wire ke along point karti ho. Yeh step kyun? Current ek scalar hai: charge ek counted total ki tarah conserve hoti hai (d t d Q
Worked example Example 4 — Forces at an angleDo forces, 6 8 9 0 ∘ = 6 2 + 8 2 + 2 ⋅ 6 ⋅ 8 cos 9 0 ∘ = 36 + 64 + 0 = 10 Yeh step kyun? Force ek vector hai; θ = 9 0 ∘ 36 + 64 θ = 0 ∘ cos 0 ∘ = 1 36 + 64 + 96 = 14 = 6 + 8 ho sakte hain — lekin sirf tab jab aligned hon.
Recall Feynman: ek 12-saal ke bacche ko explain karoPhysics mein kuch words aise hote hain jaise "mere paas 5 apples hain" — number sab kuch bata deta hai. Woh scalars hain. Doosre words aise hote hain jaise "hawa 5 km/h chal rahi hai" — yeh tab tak bekar hai jab tak main yeh na bataaun ki kis taraf chal rahi hai. Woh vectors hain.
Yeh raha magic test: agar main tumhe 3 qadam push karoon aur phir 3 qadam aur, toh kya tum hamesha 6 qadam door ho? Agar hamesha ho, toh yeh scalar hai. Lekin agar main tumhe 3 qadam daayein aur 3 qadam aage push karoon, toh tum start se 6 se kam qadam door ho (tum diagonal jaate ho!) — woh "6 se kam" wala surprise matlab hai ki direction in charge hai, isliye yeh vector hai.
"S calar = S ingle number. V ector = V alue + V ector (direction)."
Aur trap ke liye: "Ek nadi mein current hai aur direction bhi hai, lekin do nadiyon ki currents bas add ho jaati hain — current ek scalar hai." (Direction ≠ vector; adding rule decide karta hai.)
Scalar quantity ko kya define karta hai? Woh quantity jo sirf magnitude + unit se fully specify ho jaaye, ordinary arithmetic follow kare. Jaise mass, time, temperature.
Vector quantity ko kya define karta hai? Woh quantity jise magnitude AUR direction dono chahiye, jo triangle/parallelogram law se add ho. Jaise displacement, force.
"Direction hai" vector ke liye sufficient test kyun NAHI hai? Kyunki kuch directed quantities (jaise electric current) arithmetically add hoti hain, parallelogram law se nahi. Asli test addition rule hai.
Distance ka displacement se wahi rishta hai jo speed ka ___ se hai? velocity (scalar : vector pairing).
Do vectors A, B ke beech angle θ par resultant ki magnitude do. Do perpendicular forces 6 N aur 8 N — resultant? 10 N, kyunki
cos 9 0 ∘ = 0 ⇒ 36 + 64 .
Ek poore circular lap par, average speed aur average velocity kya hoti hai? Speed = total distance/time (nonzero); velocity = 0 (displacement zero hota hai).
Vector aur scalar ke liye negative sign ka kya matlab hai? Scalar: zero se kam value; Vector: same magnitude opposite direction mein.
Electric current scalar hai ya vector? Scalar (ismein direction hoti hai lekin Kirchhoff's law ke according arithmetically add hoti hai).
Temperature scalar hai ya vector? Scalar — sirf ek number aur unit se poori tarah diya jaata hai, koi direction nahi.
Vector Addition — Triangle & Parallelogram Law — A 2 + B 2 + 2 A B cos θ Distance vs Displacement — canonical scalar/vector pair.Speed vs Velocity — doosra canonical pair, kinematics se linked.Components of a Vector — vectors kaise scalar parts mein split hote hain.Dot and Cross Products — woh operations jo vectors ko waapas scalars/vectors mein turn karte hain.Units and Dimensions — har scalar/vector par phir bhi ek unit hoti hai.
Parallelogram / triangle law
Walking 3 m then 4 m problem
mass, time, speed, energy
displacement, velocity, force
Electric current has direction
Real test is addition rule
R equals sqrt A2 plus B2 plus 2AB cos theta
Direction alone means vector myth
