Equation of a line in 3D — vector, symmetric, parametric forms

3.6.6 · Maths › 3D Geometry

1. Vector form — master equation

KYUN yeh kaam karta hai (scratch se derivation): Maano $P$ line par koi bhi point hai aur $A$ uس par ek fixed point hai. Vector $A P$ line ke saath saath rehta hai, isliye yeh direction $b$ ke parallel hona chahiye. Do parallel vectors ek doosre ke scalar multiples hote hain: $A P = λ b .$ Ab $A P$ ko origin $O$ se position vectors use karke likho: $A P = O P - O A = r - a .$ Substitute karo: $r - a = λ b ⟹ r = a + λ b .$ Yeh step kyun? Kyunki "line par hona" ka matlab yehi hai ki ek fixed point se displacement, direction ka ek multiple ho — aur kuch nahi.

Figure — Equation of a line in 3D — vector, symmetric, parametric forms

2. Parametric form — vector ko coordinates mein tod do

Maano $a = (x_{1}, y_{1}, z_{1})$ aur $b = (a, b, c)$ (direction ratios). Likho $r = (x, y, z)$ .

(x,y,z)=(x_1+\lambda a,\; y_1+\lambda b,\; z_1+\lambda c)$$ > [!formula] Parametric equations > $$x = x_1+\lambda a,\qquad y=y_1+\lambda b,\qquad z=z_1+\lambda c$$ > Numbers $a,b,c$ line ke ==direction ratios== (DRs) hain. **KAISE use karein:** $\lambda$ ki koi value plug karo → ek point milega. $\lambda=0$ anchor $A$ deta hai; $\lambda=1$, $A+\vec{b}$ deta hai, wagera. --- ## 3. Symmetric (Cartesian) form — $\lambda$ hatao > [!intuition] KYUN $\lambda$ hatate hain > $\lambda$ "scaffolding" hai. Agar hum har parametric equation ko $\lambda$ ke liye solve karein, toh teeno expressions usi $\lambda$ ke barabar honge, toh woh ek doosre ke barabar hain — aur $\lambda$ gayab ho jaata hai. Parametric equations se (maano $a,b,c\neq 0$): $$\lambda=\frac{x-x_1}{a},\qquad \lambda=\frac{y-y_1}{b},\qquad \lambda=\frac{z-z_1}{c}.$$ Inhe barabar karo: > [!formula] Symmetric form > $$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\;(=\lambda)$$ > **Point** $(x_1,y_1,z_1)$ numerators se padho, **direction ratios** $(a,b,c)$ denominators se. > [!mistake] Jab koi denominator zero ho > Galat instinct: "$\frac{z-z_1}{0}$ likho aur aage badho." Yeh theek lagta hai kyunki pattern symmetric dikhta hai — lekin 0 se divide karna meaningless hai. > **Fix:** zero DR ka matlab hai line us coordinate mein **constant** hai. Agar $c=0$, likho > $$\frac{x-x_1}{a}=\frac{y-y_1}{b},\qquad z=z_1.$$ > Teesra coordinate frozen hai; yeh line ke saath kabhi nahi badlega. --- ## 4. Do points se line > [!definition] Two-point form > Points $A(\vec{a})$ aur $B(\vec{b})$ se hone wali line ki direction $\vec{b}-\vec{a}$ hai, toh > $$\vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a}).$$ > Cartesian: > $$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}.$$ **KYUN:** "$A$ se $B$ ki taraf" direction *hi* $\vec{AB}=\vec{b}-\vec{a}$ hai. Anchor bas $A$ hai. Wahi master equation, naya costume. --- ## 5. Worked examples > [!example] Example 1 — Teeno forms banao > $A(2,-1,3)$ se $\vec{b}=2\hat i+3\hat j-\hat k$ ke parallel line ke vector, parametric aur symmetric forms dhundho. > > **Vector:** $\vec{r}=(2\hat i-\hat j+3\hat k)+\lambda(2\hat i+3\hat j-\hat k)$. > *Kyun?* Master equation: anchor $+$ $\lambda\cdot$direction. > > **Parametric:** $x=2+2\lambda,\; y=-1+3\lambda,\; z=3-\lambda$. > *Kyun?* $\vec{r}$ ke components alag alag padho. > > **Symmetric:** $\dfrac{x-2}{2}=\dfrac{y+1}{3}=\dfrac{z-3}{-1}$. > *Kyun?* Har parametric equation ko $\lambda$ ke liye solve karo aur barabar karo. Note karo $y-(-1)=y+1$. > [!example] Example 2 — Do points > $P(1,0,2)$ aur $Q(4,5,2)$ se line. > > Direction $=\vec{PQ}=(4-1,\,5-0,\,2-2)=(3,5,0)$. > *Yeh step kyun?* Direction points ke beech ka displacement hai. > > Kyunki $c=0$ (z-DR zero hai), line plane $z=2$ mein rehti hai: > $$\frac{x-1}{3}=\frac{y-0}{5},\qquad z=2.$$ > *Kyun?* Dono points ka $z$-coordinate $2$ hai, toh yeh badal nahi sakta. > [!example] Example 3 — Symmetric form se info padho > Diya gaya $\dfrac{x+1}{4}=\dfrac{y-2}{-1}=\dfrac{z}{3}$, ek point aur direction batao. > > $\dfrac{x-x_1}{a}$ se match karo: numerator $x+1=x-(-1)\Rightarrow x_1=-1$. Similarly $y_1=2$, aur $z=z-0\Rightarrow z_1=0$. > **Point** $(-1,2,0)$, **direction** $(4,-1,3)$. > *Kyun?* Numerators point encode karte hain, denominators DRs — hamesha signs check karo $x-x_1$ likhke. > [!example] Example 4 — Pehle Andaza, Phir Verify > **Andaza:** Kya point $(6,1,0)$ line $\vec r=(0,3,2)+\lambda(2,-1,-1)$ par hai? Pehle guess karo. > **Verify:** $x$ solve karo: $6=0+2\lambda\Rightarrow\lambda=3$. $y$ check karo: $3+3(-1)=0\neq 1$. ✗ > Toh point line par **nahi** hai (ek $\lambda$ *teeno* equations ek saath satisfy karna chahiye). *Kyun?* Wahi parameter sabhi coordinates control karta hai. --- > [!mistake] Teen classic traps ko steel-man karo > 1. **"Direction ratios actual point displacements hone chahiye."** Sahi lagta hai kyunki two-point form mein $(x_2-x_1)$ etc. use hota hai. Lekin direction ka *koi bhi* scalar multiple kaam karta hai — $(2,3,-1)$ aur $(4,6,-2)$ ek hi line dete hain. **Fix:** DRs sirf ek nonzero scalar tak unique hain. > 2. **"Symmetric form ke numerators direction hain."** Galat — numerators **point** dete hain, denominators **direction** dete hain. **Fix:** hamesha $\frac{x-x_1}{a}$ likhke dono ko label karo. > 3. **"Ek point line par hai agar woh ek equation fit kare."** Sufficient lagta hai. **Fix:** *usi* $\lambda$ ko teeno satisfy karne chahiye; har jagah check karo. --- > [!recall]- Feynman: ek 12-saal ke bachche ko samjhao (click to reveal) > Socho tum ek bade kamre mein ek jagah khade ho aur apna haath kisi direction mein pointing kar rahe ho. Line woh **har jagah hai jo tum apne haath ke saath seedha chalne par hit karte**, aage ya peeche, kitne bhi steps. Jis jagah se tum shuru ho woh "point" hai, tumhare haath ki direction "direction" hai. Agar tum $\lambda$ steps loge (negative = peeche), tum line par utaroge. Yahi poora secret hai — starting point plus ek direction. > [!mnemonic] Structure yaad karo > **"PADDLE"** → **P**oint **A**nchor, **D**irection **D**rives, **L**ambda **E**xplores. > Aur symmetric form ke liye: **"Top = tum KAHAN ho (point), Bottom = tum KAHAN jaate ho (direction)."** --- ## #flashcards/maths 3D mein ek line uniquely determine karne ke liye kaunse do data chahiye? ::: Line par ek point aur uss ke parallel ek direction vector. $\vec a$ se $\vec b$ ke parallel line ki vector equation? ::: $\vec r = \vec a + \lambda\vec b$, $\lambda\in\mathbb R$. $\vec r-\vec a=\lambda\vec b$ kyun hai? ::: Kyunki $\vec{AP}$ line ke saath rehta hai isliye woh $\vec b$ ke parallel hai, hence uska scalar multiple hai. Symmetric form mein numerators kya encode karte hain? ::: Line par fixed point $(x_1,y_1,z_1)$. Symmetric form mein denominators kya encode karte hain? ::: Line ke direction ratios $(a,b,c)$. Zero direction ratio ko, jaise $c=0$, kaise handle karte hain? ::: Us fraction ko drop karo aur $z=z_1$ likho (coordinate constant hai) doosre equal ratios ke saath. Points $A$ aur $B$ se hone wali line ki direction? ::: $\vec b-\vec a$ (yaani $\vec{AB}$). Kya direction ratios unique hain? ::: Nahi — sirf ek nonzero scalar multiple tak; $(2,3,-1)$ aur $(4,6,-2)$ ek hi direction describe karte hain. Test kaise karein ki koi point parametric line par hai? ::: Ek coordinate se $\lambda$ dhundho aur check karo ki wahi $\lambda$ teeno equations satisfy karta hai. $\lambda=0$ par kaunsa point milta hai? ::: Anchor point $\vec a$ khud. --- ## Connections - [[Vectors — scalar multiplication & parallel vectors]] ($\vec{AP}=\lambda\vec b$ ke peeche ka engine) - [[Direction ratios and direction cosines]] - [[Angle between two lines in 3D]] - [[Shortest distance between two skew lines]] - [[Equation of a plane in 3D]] - [[Coplanarity of two lines]] - [[Distance of a point from a line in 3D]] ## 🖼️ Concept Map ```mermaid flowchart TD A[Line in 3D] -->|needs| PT[Anchor point A] A -->|needs| DIR[Direction vector b] PT -->|combine| VEC[Vector form r = a + lambda b] DIR -->|combine| VEC VEC -->|derived from| AP[AP parallel to b] VEC -->|read coordinates| PAR[Parametric form] PAR -->|x1 y1 z1| POINT[Point coords] PAR -->|a b c| DR[Direction ratios] PAR -->|eliminate lambda| SYM[Symmetric form] DR -->|denominators| SYM POINT -->|numerators| SYM VEC -->|lambda is| SCAL[Scalar parameter] ``` ## 🔬 Deep Dive > [!intuition] Aur gehraai mein jao — visual, zero se > Is topic ke step-by-step 3Blue1Brown-style breakdowns. - [[3.6.06 D1 Foundations|D1 · Foundations — har symbol zero se]] - [[3.6.06 D2 Visual Walkthrough|D2 · Visual walkthrough — derivation pictures mein]] - [[3.6.06 D3 Worked Examples|D3 · Worked examples — har scenario]] - [[3.6.06 D4 Exercises|D4 · Exercises — graded, full solutions]] - [[3.6.06 D5 Question Bank|D5 · Question bank — concept traps]]