Lines and planes in 3D — vector equations — Hinglish notes

4.5.5 · Maths › Linear Algebra (Full)

HUM vector equations kyun use karte hain?

2D mein tum ek line ko $y = m x + c$ se describe kar sakte ho. 3D mein yeh toot jaata hai: 3D space mein ek single equation $y = m x + c$ ek plane banati hai, line nahi. Hume ek aisi language chahiye jo:

kisi bhi number of dimensions mein kaam kare,
position (hum kahan se shuru karte hain) aur direction (hum kahan jaate hain) ko alag rakhe,
zero se divide kiye bina vertical/weird-slope objects ko handle kare.

Vectors exactly yahi karte hain. Ek point hume position deta hai; ek direction vector hume travel deta hai.

1. Line

HUM ise kaise derive karte hain (scratch se). Line par koi bhi point $P$ (position $r$ ) lo aur fixed point $A$ (position $a$ ) lo. Vector $A P = r - a$ line ke saath lie karta hai, isliye yeh direction $d$ ke parallel hona chahiye. Do parallel vectors scalar multiples hote hain: $r - a = t d \Rightarrow r = a + t d .$ Yeh step kyun? "Line ke saath lie karna" ek line ka poora geometric meaning hai — uske har displacement usi direction ka stretch/shrink hota hai.

Do points se guzarni wali line

$A (a)$ aur $B (b)$ se guzarni wali: direction $d = b - a$ hai, isliye $r = a + t (b - a) = (1 - t) a + t b .$ $t = 0$ par tum $A$ par ho, $t = 1$ par $B$ par, $0 \leq t \leq 1$ segment trace karta hai.

2. Plane

Ek plane ko do non-parallel directions chahiye, ya — zyada clean — ek perpendicular direction (uska normal).

Normal form kyun kaam karta hai. Plane mein kisi bhi point $P$ ke liye, $r - a$ plane ke andar lie karta hai, isliye normal $n$ ke perpendicular hota hai. Perpendicular ⇒ dot product zero. Yeh single equation plane ke har point ko capture karta hai.

Parametric se normal par kaise jaayein: normal dono spanning directions ke perpendicular hona chahiye, isliye $n = u \times v .$ Yeh step kyun? Cross product exactly woh vector produce karta hai jo dono inputs ke orthogonal ho — yeh plane ke normal ki geometric definition hai.

Figure — Lines and planes in 3D — vector equations

3. Distances (80/20 high-value formulas)

Worked Examples

Common mistake Steel-manned errors

(a) "Alag parameter $t$ values ⇒ alag lines." Kyun sahi lagta hai: numbers badalne se usually objects badal jaate hain. Fix: ek line points ka ek set hai; kaafi saare $(a, d)$ pairs usi set ko describe karte hain. Lines equal hain iff directions parallel hain aur woh ek point share karte hain.

(b) "Cartesian plane $a x + b y + cz = d$ — normal $(a, b, c, d)$ hai." Kyun sahi lagta hai: charon numbers symmetric lagte hain. Fix: normal sirf $(a, b, c)$ hai. Constant $d$ plane ko $n$ ke saath shift karta hai; yeh direction ka part nahi hai.

(c) Jab ek denominator 0 ho tab symmetric form use karna. Fix: kabhi $\frac{z - z _{0}}{0}$ mat likho. Agar $c = 0$ hai, $z = z_{0}$ state karo aur symmetric form sirf baaki coordinates par use karo.

(d) Line ki "direction" aur plane ke "normal" ko confuse karna. Line ka vector uske saath point karta hai; plane ka normal uske paas se point karta hai. Inhe mix karne se parallel ↔ perpendicular flip ho jaata hai.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum ek giant gym mein khade ho. Line matlab hai "yahan khado, phir ek arrow ki direction mein chalta raho" — tum sirf us arrow ke saath aage ya peeche ja sakte ho. Plane matlab hai poora floor: jahan tum khade ho wahan se tum do arrow directions mein slide kar sakte ho aur flat jagah kisi bhi jagah pahunch sakte ho. Floor ko jaldi describe karne ke liye, do slide-directions dene ki jagah, tum ek arrow dete ho jo floor se seedha upar point karta hai (yeh normal hai). Koi bhi point "floor par" tabhi hai jab us tak jaane mein us up-arrow ke relative kabhi upar ya neeche na jao — exactly yahi dot product ka zero hona hai.

Connections

Vectors and scalar (dot) product — normal-form aur projections ko power karta hai.
Cross product — do directions se plane ka normal banata hai.
Vector projection — dono distance formulas ka engine.
Intersection of lines and planes — parametric equations ko simultaneously solve karo.
Systems of linear equations — ek Cartesian plane ek linear equation hai; intersections = systems solve karna.
Parametrisation and parameters — $t, s$ ka role.

Vector equation of a line through

a

with direction

d

r = a + t d, t \in R

Direction vector of the line through points

A, B

b - a

Parametric form of

r = a + t d

with

a = (x_{0}, y_{0}, z_{0}), d = (a, b, c)

x = x_{0} + t a, y = y_{0} + t b, z = z_{0} + t c

Symmetric form of a line

\frac{x - x _{0}}{a} = \frac{y - y _{0}}{b} = \frac{z - z _{0}}{c}

(each denominator nonzero)

Parametric (two-direction) equation of a plane

r = a + s u + t v

Normal (scalar) form of a plane

(r - a) \cdot n = 0

, i.e.

r \cdot n = d

How to get a plane's normal from two spanning directions

n = u \times v

What are the coefficients in

a x + b y + cz = d

normal vector

(a, b, c)

ke components

Distance from point

p

to plane

r \cdot n = d

\frac{∣ p \cdot n - d ∣}{∣ n ∣}

Distance from point

p

to line

r = a + t d

\frac{∣ ( p - a ) \times d ∣}{∣ d ∣}

Condition for two lines to be identical

directions parallel HON AUR ek common point share karen

What does

t

represent in

r = a + t d

ek scalar parameter jo decide karta hai ki line par kaunsa point hai

Lines and planes in 3D — vector equations