Angle between two lines

3.6.7 · Maths › 3D Geometry

HUM MEASURE KYA KAR RAHE HAIN?

ACUTE angle kyun lete hain? Do intersecting lines asliyat mein do pairs of angles banati hain ( $θ$ aur $18 0^{\circ} - θ$ ). Ek line "dono taraf" point karti hai, toh ek direction vector $b$ ko $- b$ mein flip karne par wahi line milti hai lekin supplementary angle milta hai. Ek single, unambiguous answer paane ke liye hum formula mein modulus $∣ \cdot ∣$ lagate hain taaki angle acute rahe.

KAISE: formula scratch se derive karo

Hum sirf ek cheez se shuru karte hain — dot product definition.

Step 1 — Dot product definition. $b_{1} \cdot b_{2} = ∣ b_{1} ∣ ∣ b_{2} ∣ cos θ$ Yeh step kyun? Yeh algebraic operation (components) aur geometric quantity (angle) ke beech ka first-principles bridge hai.

Step 2 — $cos θ$ ke liye solve karo. $cos θ = \frac{b _{1} \cdot b _{2}}{∣ b _{1} ∣ ∣ b _{2} ∣}$ Yeh step kyun? Hum angle chahte hain, toh $cos θ$ isolate karo.

Step 3 — Acute angle force karo. Kyunki $b$ aur $- b$ same line dete hain, hum numerator ki absolute value lete hain: $cos θ = \frac{∣ b _{1} \cdot b _{2} ∣}{∣ b _{1} ∣ ∣ b _{2} ∣}$ Yeh step kyun? $∣ cos θ ∣$ guarantee karta hai ki $θ \in [0^{\circ}, 9 0^{\circ}]$ .

Ek clean $sin$ form bhi hai cross product use karke: $sin θ = \frac{∣ b _{1} \times b _{2} ∣}{∣ b _{1} ∣ ∣ b _{2} ∣}$ Yeh useful kyun hai? Kabhi kabhi cross product haath mein hota hai; saath hi $tan θ$ safely nikalne deta hai.

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: ek 12 saal ke bachche ko samjhao

Socho do pencils ek table par rakhi hain, dono ek hi jagah se guzar rahi hain. Jahan woh milti hain wahan ek angle banta hai — aur woh angle iss baat ki parwah nahi karta ki pencils table par kahan hain, sirf iss baat ki ki woh kis taraf point karti hain. Ek "direction arrow" batata hai ki ek pencil kis taraf point karti hai. Do arrows ke beech ka angle nikalne ke liye hum ek magic recipe use karte hain jise dot product kehte hain: matching parts ko multiply karo, add karo, phir arrows ki lengths se divide karo. Result angle ka cosine hota hai. Hum answer ko "chota" bhi rakhte hain (hamesha right angle se kam) kyunki baayi taraf point karne wali pencil wahi line hai jo daayein taraf point karne wali pencil hai.

Active-recall flashcards

3D mein do lines ke beech ka angle kya determine karta hai?

Sirf unke direction vectors (orientation), unki position nahi.

Direction ratios wali lines ke beech ke angle ka formula?

cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}

Formula mein modulus kyun hai?

Kyunki lines ke beech ka angle acute define hota hai; ek line dono taraf point karti hai, toh

- b

supplement deta hai.

Do lines ke perpendicular hone ki condition?

a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0

(dot product zero).

Do lines ke parallel hone ki condition?

Direction ratios proportional:

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

Direction cosines ke saath

cos θ

kya hota hai?

∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣

(denominator 1 hai, kyunki woh unit vectors hain).

Symmetric form

\frac{x - x _{0}}{a} = \dots

mein direction ratios kaun se numbers hain?

Denominators

(a, b, c)

— point

(x_{0}, y_{0}, z_{0})

nahi.

Angle ka cross-product form?

sin θ = \frac{∣ b _{1} \times b _{2} ∣}{∣ b _{1} ∣∣ b _{2} ∣}

(1, 2, 2)

aur

(2, 2, 1)

ke beech ka angle?

cos^{- 1} (8/9) \approx 27. 3^{\circ}

Connections

Direction cosines and direction ratios — is formula ke inputs.
Dot product of vectors — derivation ka engine.
Cross product of vectors — $sin θ$ form deta hai.
Equation of a line in 3D — jahan se hum direction ratios padhte hain.
Angle between two planes — directions ki jagah normals use karta hai, same dot-product idea.
Angle between a line and a plane — plane ke normal ke saath $sin θ$ use karta hai.
Shortest distance between two lines — yeh bhi direction-vector based hai.