Relation between direction cosines - l² + m² + n² = 1

3.6.5 · Maths › 3D Geometry

WHAT are direction cosines?

HOW to derive $l^{2} + m^{2} + n^{2} = 1$ (bilkul scratch se)

Step 1 — Unit vector set up karo. Line ke along ek unit vector $\overset{u}{^}$ lo. Usse components mein likho: $\overset{u}{^} = (u_{x}, u_{y}, u_{z}) .$

Yeh step kyun? Direction ki apni koi length nahi hoti, isliye hum usse length $1$ par standardise karte hain. Isse "kitna door" hat jaata hai, sirf "kaunsi disha" bachti hai.

Step 2 — Har component ko direction cosine ki tarah identify karo. $\overset{u}{^}$ ka $x$ -component $x$ -axis par uska projection hai: $u_{x} = \overset{u}{^} \cdot \hat{i} = cos α = l .$ Isi tarah $u_{y} = cos β = m$ aur $u_{z} = cos γ = n$ .

Yeh step kyun? Ek unit vector ka kisi axis par projection hi us axis ke saath banaye angle ka cosine hota hai — yahi dot product ka geometric matlab hai.

Step 3 — Use karo ki $\overset{u}{^}$ ki length 1 hai. $∣ \overset{u}{^} ∣^{2} = u_{x}^{2} + u_{y}^{2} + u_{z}^{2} = 1.$

Yeh step kyun? Yeh sirf 3D Pythagoras theorem (distance formula) hai jo ek unit vector par apply hua hai.

Step 4 — Substitute karo. $l^{2} + m^{2} + n^{2} = 1$

Figure — Relation between direction cosines - l² + m² + n² = 1

From Direction Ratios to Direction Cosines

Agar $r = (a, b, c)$ hai aur magnitude $a^{2} + b^{2} + c^{2}$ hai, to $l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \frac{c}{a ^{2} + b ^{2} + c ^{2}} .$

Kyun? Kisi bhi vector ko uski length se divide karne par hamesha ek unit vector banta hai, jiske components automatically $l^{2} + m^{2} + n^{2} = 1$ satisfy karte hain.

Common Mistakes

Active Recall

Recall Quick self-test (chhupaao aur jawab do)

Direction cosines ke beech relation batao. → $l^{2} + m^{2} + n^{2} = 1$ .
Yeh exactly 1 kyun hota hai? → Kyunki $l, m, n$ ek unit vector ke components hain.
Ratios $(a, b, c)$ se cosines kaise nikaalte hain? → Har ek ko $a^{2} + b^{2} + c^{2}$ se divide karo.
$sin^{2} α + sin^{2} β + sin^{2} γ$ kya hota hai? → $2$ .
Kitne direction cosines independent hote hain? → Do.

Recall Feynman: ek 12 saal ke bachche ko samjhao

Socho ek seedha teer ek kamre mein hawa mein tair raha hai. Apne dost ko batane ke liye ki yeh bilkul kaunsi disha mein point kar raha hai, tum puchho: "Yeh daayein waali deewar ki taraf kitna jhuka hai? Aage waali deewar ki taraf? Chhat ki taraf?" Har "jhukav" ek number hai $- 1$ aur $1$ ke beech. Yahan magic hai: agar teeno jhukavon ko square karo aur jodo, to HAMESHA exactly $1$ milta hai — chahe teer kisi bhi disha mein point kare! Kyunki teer ki length "ek kadam" ki fixed hai, aur yeh teen jhukav yehi batate hain ki us ek kadam ka kitna hissa sideways, aage, aur upar gaya. Pythagoras ke rule se, yeh tukde milke poora kadam banana chahiye: $1$ .

80/20 — Woh 20% jo tumhe zaroor yaad rakhna hai

Direction cosines = unit direction vector ke components.
$\Rightarrow l^{2} + m^{2} + n^{2} = 1$ (unit vector par Pythagoras).
Ratios → cosines: magnitude se divide karo.
Hamesha $\pm$ likho; sirf 2 independent hote hain.

Connections

Direction Ratios — direction cosines ke unnormalised cousins.
Unit Vectors — identity ka geometric source.
Dot Product — kyun projection = cosine hota hai.
Distance Formula in 3D — Pythagoras step deta hai.
Angle Between Two Lines — $cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ use karta hai.
Equation of a Line in 3D — direction cosines line ka slope define karte hain.

Ek line ke direction cosines ke beech kya relation hai?

l^{2} + m^{2} + n^{2} = 1

Direction cosines ke squares ka sum 1 kyun hota hai?

Kyunki

l, m, n

line ke saath ek unit vector ke components hain, aur ek unit vector ki magnitude 1 hoti hai (3D mein Pythagoras).

Direction ratios

(a, b, c)

ko direction cosines mein kaise convert karte hain?

Har ek ko magnitude se divide karo:

l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}

, similarly

m, n

ke liye.

Direction angles ke liye

sin^{2} α + sin^{2} β + sin^{2} γ

kya hoga?

2

(kyunki

cos^{2} α + cos^{2} β + cos^{2} γ = 1

aur 3 mein se subtract karo).

Teen direction cosines mein se kitne independent hote hain?

Do — teesra sign tak

n = \pm 1 - l^{2} - m^{2}

se fixed hota hai.

Agar

l = \frac{1}{2}, m = \frac{1}{2}

ho, to

n

nikalo.

n^{2} = 1 - \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \Rightarrow n = \pm \frac{1}{2}

Kya koi line teeno axes se

3 0^{\circ}

ka angle bana sakti hai?

Nahi, kyunki

3 cos^{2} 3 0^{\circ} = 2.25 \neq = 1

(1, 2, 3)

aur

(4, 6, 3)

se hoke jaane wali line ke direction cosines?

Ratios

(3, 4, 0)

, magnitude 5, to

(\frac{3}{5}, \frac{4}{5}, 0)

Direction ratios generally

a^{2} + b^{2} + c^{2} = 1

kyun satisfy nahi karte?

Yeh normalised nahi hote; sirf length se divide karne ke baad components ek unit vector banate hain.