Skew lines — shortest distance — Hinglish notes

3.6.12 · Maths › 3D Geometry

HUM KYA compute kar rahe hain?

KAISE: First principles se derivation

Hum chahte hain wo sabse chhota segment jo $L_{1}$ par point $P$ ko $L_{2}$ par point $Q$ se jode.

Step 1 — Geometric insight (WHY perpendicular?). Maano $P Q$ joining vector hai. Agar $P Q$ $L_{1}$ par perpendicular nahi hota, toh hum $P$ ko $L_{1}$ par thoda sa slide kar sakte the aur $P Q$ ko chhota kar sakte the (ek perpendicular drop karo — hypotenuse hamesha leg se bada hota hai). Toh minimum par, $P Q ⊥ b_{1}$ . Usi argument se $P Q ⊥ b_{2}$ . Yeh step kyun? Distance ka minimum wahan hota hai jahan kisi bhi line par move karne se first-order change nahi aata — yahi exactly perpendicularity hai.

Step 2 — Common perpendicular ki direction. Ek vector jo dono $b_{1}$ aur $b_{2}$ par perpendicular ho, use $n = b_{1} \times b_{2} .$ ki direction mein hona chahiye. Yeh step kyun? Cross product by definition dono inputs par perpendicular hota hai. (Yeh nonzero tab hota hai jab lines parallel nahi hoti — yahi woh jagah hai jahan "skew ke liye $b_{1} ∦ b_{2}$ " use hota hai.)

Step 3 — Gap ko $n$ par project karo. Har line par KOI BHI point lo: $a_{1}$ aur $a_{2}$ . Unke beech ka vector hai $a_{2} - a_{1}$ . Shortest distance is gap ke common perpendicular direction $\overset{n}{^} = n /∣ n ∣$ ke along component ki length hai: $d = ∣ (a_{2} - a_{1}) \cdot \overset{n}{^} ∣ = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣} .$ Yeh step kyun? Chahe hum $a_{1}, a_{2}$ koi bhi points chunein, along- $\overset{n}{^}$ component same rehta hai, kyunki kisi line par move karna sirf $b$ -vectors add karta hai, jo $⊥ \overset{n}{^}$ hain aur kuch contribute nahi karte.

Worked examples

Cartesian form

$\frac{x - x _{1}}{a _{1}} = \frac{y - y _{1}}{b _{1}} = \frac{z - z _{1}}{c _{1}}$ aur $\frac{x - x _{2}}{a _{2}} = \frac{y - y _{2}}{b _{2}} = \frac{z - z _{2}}{c _{2}}$ ke liye: $d = \frac{x _{2} - x _{1} a _{1} a _{2} y _{2} - y _{1} b _{1} b _{2} z _{2} - z _{1} c _{1} c _{2}}{( b _{1} c _{2} - b _{2} c _{1} ) ^{2} + ( c _{1} a _{2} - c _{2} a _{1} ) ^{2} + ( a _{1} b _{2} - a _{2} b _{1} ) ^{2}} .$ Same formula, bas components ke saath likha hua.

Common mistake Common errors ko pakdo

(1) $a_{2} - a_{1}$ mein galat points use karna... phir panic karna. Kyun sahi lagta hai: "Zaroor answer change hoga agar main line par alag point se shuru karun." Fix: Nahi hoga — gap mein $b$ ka multiple add karna $\overset{n}{^}$ ke saath dot karne par kuch nahi karta (kyunki $b \cdot \overset{n}{^} = 0$ ). Tum koi bhi convenient points chun sakte ho.

(2) Absolute value bhool jaana / galat magnitude se divide karna. Kyun sahi lagta hai: triple product final answer jaisa lagta hai. Fix: distance $\geq 0$ ⇒ $∣ \cdot ∣$ lo, aur $∣ b_{1} \times b_{2} ∣$ se divide karo, $∣ b_{1} ∣∣ b_{2} ∣$ se NAHI.

(3) Parallel lines par cross-product formula apply karna. Kyun sahi lagta hai: yeh "the formula" hai. Fix: $b_{1} \times b_{2} = 0$ → $0/0$ . Parallel-line formula par switch karo.

(4) Sirf isliye skew conclude karna kyunki lines ka koi obvious common point nahi dikhta. Fix: triple product compute karo. $= 0$ (aur parallel nahi) ⇒ wo intersect karti hain, skew nahi.

Recall Feynman: ek 12-saal ke bachhe ko samjhao

Socho do seedhi wire ki tukdein ek room mein float kar rahi hain jo kabhi nahi milti aur same direction mein nahi hain. Exactly ek sabse chhoti stick hai jo tum unhe connect karne ke liye rakh sakte ho — aur woh stick dono wires ko perfect right angle par chhuegi. Uski length nikalne ke liye: har wire par ek point chuno, unke beech ek arrow banao, phir sirf woh part measure karo jo "dono par right-angle" direction mein point karta hai. Cross product tumhe woh direction batata hai; dot product measure karta hai ki arrow ka kitna hissa uss taraf point karta hai.

Flashcards

Do lines skew kab hoti hain?

Na parallel na intersecting (alag directions AUR koi common point nahi).

Skew lines ke liye shortest-distance formula?

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

Shortest segment dono lines par perpendicular kyun hota hai?

Warna tum ek point ko line par slide karke use chhota kar sakte; min distance ⇒ koi first-order change nahi ⇒ perpendicular.

Cross product kyun aata hai?

b_{1} \times b_{2}

by definition dono direction vectors par perpendicular hai — common-perpendicular direction.

Numerator

= 0

(lines parallel nahi) ka kya matlab hai?

Lines coplanar hain aur isliye intersect karti hain.

a_{1}, a_{2}

ke liye koi bhi point kyun chun sakte hain?

b

ka multiple gap mein add karna

⊥ \overset{n}{^}

hai, isliye

\overset{n}{^}

par projection change nahi hota.

PARALLEL lines ke beech shortest distance ka formula?

d = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣}

Numerator scalar triple product geometrically kya represent karta hai?

a_{2} - a_{1}, b_{1}, b_{2}

se bane parallelepiped ka signed volume.

Connections

Vector Cross Product — perpendicular direction deta hai.
Scalar Triple Product — numerator; zero ⇔ coplanar.
Distance from a Point to a Line (3D) — parallel-line case.
Equation of a Line in 3D — vector & Cartesian forms.
Coplanarity of Two Lines — condition $[a_{2} - a_{1}, b_{1}, b_{2}] = 0$ .
Plane containing a line — common perpendicular related planes mein hoti hai.

Skew lines — shortest distance