3.6.4
3D mein ek line kisi direction mein point karti hai. Us direction ko numbers se capture karne ke liye hum poochte hain:
"Yeh line x y z angles jo line har axis ke saath banati hai.
Direction cosines un teeno angles ke cosines hain, aur direction ratios koi bhi aise numbers hain jo unke proportional hon.
WHY cosine aur angle khud kyun nahi? Kyunki cosines khoobsurati se combine hote hain — yeh exactly ek unit vector ke components hote hain line ke along. Components vectors ki tarah add aur dot karte hain; raw angles nahi karte.
Definition Direction anglesAgar ek directed line positive x y z α , β , γ α , β , γ direction angles hain.
Definition Direction cosines (DCs)Direction cosines hain
l = cos α , m = cos β , n = cos γ . ( l , m , n )
WHY yeh special hain: Ek unit vector u ^ = ( u 1 , u 2 , u 3 ) u ^ x i ^ = ( 1 , 0 , 0 ) cos α = ∣ u ^ ∣∣ i ^ ∣ u ^ ⋅ i ^ = u 1 . l = u 1 m = u 2 , n = u 3 Direction cosines hi line ke along unit vector ke components HAIN. Yahi ek fact neeche sab kuch drive karta hai.
HOW hum yeh derive karte hain (scratch se): Kyunki ( l , m , n ) = u ^ unit vector hai,
∣ u ^ ∣ 2 = l 2 + m 2 + n 2 = 1.
Common mistake Steel-man: "Surelycos 2 + cos 2 + cos 2 Galat idea kyun sahi lagta hai: 2D mein aapko yaad hoga cos 2 θ + sin 2 θ = 1 cosines ke squares ko 1 mein add hote dekhna suspicious lagta hai.
Fix: 3D mein teen axes mutually perpendicular hain, isliye teen cosines genuinely ek unit vector ke teen orthogonal components hain. Ek unit vector ke components hamesha (component1 2
Intuition Hamesha DCs kyun use nahi karte?Do points ( x 1 , y 1 , z 1 ) ( x 2 , y 2 , z 2 ) ( x 2 − x 1 , y 2 − y 1 , z 2 − z 1 ) . easy hain padhne ke liye — lekin inke squares usually 1 mein add nahi hote. Yeh DCs ke proportional hain, equal nahi. Toh hum inhe ek naam dete hain.
Definition Direction ratios (DRs)Koi bhi teen numbers a , b , c proportional hon, direction ratios kehlate hain:
l : m : n = a : b : c . infinitely many sets ho sakte hain (koi bhi nonzero scalar multiple kaam karta hai) lekin DCs ke sirf do sets hote hain (line ke saath har direction ke liye ek).
Hum chahte hain l = k a , m = k b , n = k c k l 2 + m 2 + n 2 = 1 k 2 ( a 2 + b 2 + c 2 ) = 1 ⇒ k = ± a 2 + b 2 + c 2 1 .
HOW: Har DC triple ek unit vector hai. Do unit vectors ka dot product unke beech ke angle ka cosine hota hai:
cos θ = u ^ 1 ⋅ u ^ 2 = l 1 l 2 + m 1 m 2 + n 1 n 2 .
Worked example Example 1 — Ek known vector ke DCsDirection ( 2 , − 1 , 2 )
Step 1. 2 2 + ( − 1 ) 2 + 2 2 = 9 = 3 Kyun? Yeh length DRs ko unit vector mein normalise karti hai.
Step 2. l = 3 2 , m = − 3 1 , n = 3 2 Kyun? Har ratio ko length se divide karo.
Check: 9 4 + 9 1 + 9 4 = 1
Worked example Example 2 — Do points se lineA ( 1 , 2 , 3 ) B ( 4 , 5 , 6 )
Step 1. DRs = ( 4 − 1 , 5 − 2 , 6 − 3 ) = ( 3 , 3 , 3 ) Kyun? A B Step 2. Simplify karo: ( 3 , 3 , 3 ) ∝ ( 1 , 1 , 1 ) Kyun? DRs sirf scale tak defined hain, toh sabse chota use karo.
Step 3. Length = 3 = ( 3 1 , 3 1 , 3 1 ) Kyun? Normalise karo.
Worked example Example 3 — Do lines ke beech angleDRs ( 1 , 2 , 2 ) ( 2 , 2 , 1 )
Step 1. Numerator = 1 ⋅ 2 + 2 ⋅ 2 + 2 ⋅ 1 = 2 + 4 + 2 = 8 Kyun? DRs ka dot product.
Step 2. Magnitudes = 1 + 4 + 4 = 3 4 + 4 + 1 = 3 Kyun? Angle ke cosine mein convert karne ke liye zaroori hai.
Step 3. cos θ = 9 8 ⇒ θ = cos − 1 9 8 ≈ 27. 3 ∘ Kyun? Dot product ko lengths ke product se divide karo.
Worked example Example 4 — Missing DC nikaloEk line x y α = 6 0 ∘ β = 4 5 ∘ γ
Step 1. l = cos 6 0 ∘ = 2 1 , m = cos 4 5 ∘ = 2 1 Step 2. n 2 = 1 − l 2 − m 2 = 1 − 4 1 − 2 1 = 4 1 Kyun? l 2 + m 2 + n 2 = 1 Step 3. n = ± 2 1 ⇒ γ = 6 0 ∘ 12 0 ∘ Kyun? Do valid lines hain, dono directions.
Common mistake Steel-man: "DRs aur DCs ek hi cheez hain"Kyun sahi lagta hai: Dono direction describe karte hain, aur bahut se textbook lines mein DRs already chhote hote hain.
Fix: DCs sign tak unique hote hain aur l 2 + m 2 + n 2 = 1 koi bhi proportional triple hain aur almost kabhi bhi sum-of-squares 1 nahi hota. DCs bolne se pehle hamesha normalise karo.
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho ek room mein laser pen kisi direction mein point kar rahe ho. Apne dost ko exactly batane ke liye ki yeh kahan point kar raha hai, tum batate ho ki yeh saamne wali wall (x), side wali wall (y), aur ceiling (z) ki taraf kitna jhuk raha hai. "Direction cosines" sirf teen lean-numbers hain, itne scale kiye gaye ki unhe square karke add karo toh hamesha exactly 1 mile — jaise laser arrow ki length 1 ho. "Direction ratios" lazy version hain: koi bhi teen numbers usi proportion mein, inhe length 1 tak shrink karne ki zaroorat nahi.
"DRs casual hain, DCs official hain."
DRs = Roughly proportional (koi bhi scale).
DCs = Clean kiye gaye (unit length: l 2 + m 2 + n 2 = 1 "dot the units" → lines ke beech angle sirf l 1 l 2 + m 1 m 2 + n 1 n 2
Ek line ke direction cosines kya hote hain? Cosines l = cos α , m = cos β , n = cos γ
Direction cosines kis identity ko satisfy karte hain? l 2 + m 2 + n 2 = 1
Direction ratios kya hote hain? Koi bhi teen numbers a , b , c l : m : n = a : b : c
Ek line ke DCs vs DRs ke kitne sets hote hain? Exactly do sets DCs ke (opposite directions); infinitely many sets DRs ke.
DRs ( a , b , c ) l , m , n = a 2 + b 2 + c 2 ± a , ± b , ± c .
( x 1 , y 1 , z 1 ) ( x 2 , y 2 , z 2 ) ( x 2 − x 1 , y 2 − y 1 , z 2 − z 1 )
DCs ( l 1 , m 1 , n 1 ) , ( l 2 , m 2 , n 2 ) cos θ = l 1 l 2 + m 1 m 2 + n 1 n 2
Do lines (DRs) ke perpendicular hone ki condition? a 1 a 2 + b 1 b 2 + c 1 c 2 = 0
Do lines (DRs) ke parallel hone ki condition? a 1 / a 2 = b 1 / b 2 = c 1 / c 2
cos 2 α + cos 2 β + cos 2 γ = 1 Kyunki teen axes orthogonal hain, isliye teen cosines ek unit vector ke orthogonal components hain, jiska squared length 1 hota hai.
normalise karo root a2+b2+c2 se divide karo
Direction angles alpha beta gamma
