Foundations — Lines and planes in 3D — vector equations

4.5.5 · D1 · Maths › Linear Algebra (Full) › Lines and planes in 3D — vector equations

Neeche aapko milenge symbols jaise $\mathbf{a}$ (ek position), $\mathbf{d}$ (ek direction), aur $\mathbf{r}$ (ek general, roaming point ki position). Ghabrao mat ki abhi yeh undefined hain — yeh poora note inhe ek-ek karke, ek doosre pe depend karne ke order mein, scratch se build karne ke liye hai. Last section tak aap $\mathbf{r}=\mathbf{a}+t\mathbf{d}$ aur $(\mathbf{r}-\mathbf{a})\cdot \mathbf{n}=0$ poori samajh ke padh paoge. Kuch bhi skip mat karo.

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0. Scene: 3D space aur origin

Ek khaali kamre ki imagine karo. Ek kona chuno aur use $O$ kaho, origin — woh "aap yahaan hain" dot jisse sab kuch measure hota hai. $O$ se hum teen arrows right angles par kheenchte hain: ek floor ke saath right ki taraf ($x$), ek floor ke saath aapse door ($y$), ek seedha upar ($z$). Yeh axes hain.

Hum inhe ek standard arrangement mein fix karte hain jise right-handed kehte hain: apne daayein haath ki ungliyan $x$ ke saath point karo aur unhe $y$ ki taraf curl karo; tab aapka thumb $z$ ke along point karega. Is page ki har picture — aur poora topic — yahi handedness use karta hai. (Agar aapne $z$ ko neeche point karne ke liye flip kiya, toh cross products ulte aate, isliye hum ise abhi pin kar dete hain.)

Kamre mein koi bhi jagah "$x$ steps right, $y$ steps forward, $z$ steps up" se reach hoti hai. Woh teen numbers us jagah ke coordinates hain, likhe jaate hain $(x,y,z)$.

Figure s01 dekho: red dot $P$ ko corner $O$ se teen dashed perpendicular steps se reach kiya jaata hai — pehle $x$ ke along, phir $y$, phir $z$.

Figure s01 — right-handed axes mein origin se teen perpendicular steps se ek point pin hota hai.

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1. Vector — "arrow" $\mathbf{a}$, $\mathbf{d}$, $\mathbf{r}$

Vector ek arrow hai. Iska ek length hai (kitna dur) aur ek direction (kis taraf), lekin yeh parwah nahi karta ki yeh kahaan se start hota hai — ise kahin bhi slide karo aur yeh same vector hai. Hum vectors bold mein likhte hain: $\mathbf{a}$, $\mathbf{d}$, $\mathbf{n}$, $\mathbf{r}$. Paper par aap $\vec{a}$ dekh sakte ho; same cheez hai.

3D mein ek vector uske teen step-sizes ke roop mein store hota hai: $\mathbf{d}=(a,b,c)$ matlab "$a$ along $x$, $b$ along $y$, $c$ along $z$." Yeh teen numbers iske components hain.

Teen names jo aap baar baar milenge:

• $\mathbf{a}$ — ek fixed position vector (arrow from $O$ to a chosen anchor point $A$).
• $\mathbf{d}$ — ek direction vector (ek line kis taraf jaati hai).
• $\mathbf{r}$ — ek general, roaming point ka position vector: ise "arrow from $O$ to jis bhi point ke baare mein hum abhi baat kar rahe hain" samjho. Jaise-jaise woh point line ya plane ke saath move karta hai, $\mathbf{r}$ badalta hai; $\mathbf{a}$ aur $\mathbf{d}$ wahi rehte hain.

Isliye parent likhta hai "point $A$ (position vector $\mathbf{a}$)": $A$ dot hai, $\mathbf{a}$ $O$ se arrow hai jo use reach karta hai.

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2. Adding, subtracting, aur stretching — $+$, $-$, $t\mathbf{d}$

Do vectors ko Add karna = "ek walk karo, phir doosra." $\mathbf{a}$ draw karo, phir $\mathbf{d}$ wahaan se start karo jahaan $\mathbf{a}$ khatam hua; bilkul shuru se bilkul end tak ka arrow $\mathbf{a}+\mathbf{d}$ hai. Components mein bas slot by slot add karo: $\mathbf{a}+\mathbf{d}=(a_1+d_1,a_2+d_2,a_3+d_3).$

Subtracting utni hi important hai, kyunki topic constantly $\mathbf{r}-\mathbf{a}$ likhta hai. Component-wise: $\mathbf{r}-\mathbf{a}=(r_1-a_1,r_2-a_2,r_3-a_3).$ Iska picture sabse important wala hai: $\mathbf{r}-\mathbf{a}$ woh arrow hai jo point $A$ se point $R$ tak jaata hai (woh "difference arrow," often $\overrightarrow{AR}$ likha jaata hai). Reason: $O$-to-$R$ jaane ke liye, aap pehle $O$-to-$A$ ja sakte ho (woh $\mathbf{a}$ hai) phir $A$-to-$R$; toh $\mathbf{a}+(\mathbf{r}-\mathbf{a})=\mathbf{r}$, exactly isliye bacha hua piece $\mathbf{r}-\mathbf{a}$ $A$ se $R$ ki taraf point karta hai. Jab bhi aap "$\mathbf{r}-\mathbf{a}$" dekhte ho, anchor se roaming point ko join karne wale arrow ki picture banao.

Ek vector ko number $t$ se Scale karna (ek plain number scalar kehlata hai) = "direction rakho, length ko $t$ factor se badlo." Toh $2\mathbf{d}$ do guna lamba hai; $-1\mathbf{d}$ use palat deta hai; $0\mathbf{d}$ use kuch nahi kar deta. $t\mathbf{d}=(ta,tb,tc).$

Figure s02 dono moves dikhata hai: left mein, red arrow $\mathbf{r}-\mathbf{a}$ anchor $A$ ko roaming point $R$ se join karta hai; right mein, $\mathbf{d}$ stretch hoke $2\mathbf{d}$ (red, lamba) aur flip hoke $-\mathbf{d}$ ban jaata hai.

Figure s02 — left: difference arrow $\mathbf{r}-\mathbf{a}$ $A$ se $R$ tak jaata hai; right: scaling ek arrow ko rotate kiye bina stretch ya flip karta hai.

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3. Parallel vectors aur "scalar multiple"

Do vectors parallel hote hain jab woh same way ya exactly opposite ways point karte hain — matlab ek doosre ka sirf stretched/flipped copy hai. Symbols mein: $\mathbf{u}$, $\mathbf{v}$ ke parallel hai exactly jab $\mathbf{u}=t\mathbf{v}$ kisi scalar $t$ ke liye.

Topic ko kyun chahiye: poori line derivation ek sentence par tikti hai — "$\mathbf{r}-\mathbf{a}$ (anchor se roaming point tak arrow, §2 se) line ke along lie karta hai, toh yeh $\mathbf{d}$ ke parallel hai." Parallel matlab "scalar multiple," jo deta hai $\mathbf{r}-\mathbf{a}=t\mathbf{d}$.

Recall Quick check

Agar $\mathbf{d}=(2,4,0)$ hai, toh kya $(1,2,0)$ iske parallel hai? ::: Haan — yeh $\frac{1}{2}\mathbf{d}$ ke barabar hai, ek scalar multiple.

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4. Parameter $t$ (aur $s$)

$t$ parameter hai: ek free-running dial jo aap kisi bhi real number par set kar sakte ho, likha jaata hai $t\in \mathbb{R}$ ("$t$ real numbers se belong karta hai"). $t$ ki har value aapko line par ek point deti hai. Dial ko saari values se ghuma do aur aap poori line trace karte ho.

Ek plane ko do independent dials chahiye, $s$ aur $t$, kyunki aap ek flat surface par do independent directions mein slide kar sakte ho.

General idea ke liye dekhein Parametrisation and parameters.

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5. Length (magnitude) — $|\mathbf{v}|$

Bars $|\mathbf{v}|$ matlab arrow $\mathbf{v}$ ki length. 3D mein Pythagoras se: $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}.$ Arrow ko ek box ke long diagonal ki tarah picture karo jiske sides $v_1,v_2,v_3$ hain; uski length wahi formula hai.

Topic ko kyun chahiye: ek unit vector (length exactly 1) banane ke liye hum length se divide karte hain, $\hat{\mathbf{n}}=\mathbf{n}/|\mathbf{n}|$, provided $\mathbf{n}\ne 0$ (zero-length arrow ki koi direction nahi hai aur $|\mathbf{n}|=0$, toh hum isse divide nahi kar sakte). Hat $\hat{}$ hamesha "length-1 version" mean karta hai. Parent note mein saari distances $|\mathbf{n}|$ ya $|\mathbf{d}|$ se divide hoti hain — exactly is liye legal kyunki plane ka normal aur line ka direction, by definition, nonzero hain.

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6. Dot product — $\mathbf{u}\cdot \mathbf{v}$

Dot product do vectors leta hai aur ek number return karta hai: $\mathbf{u}\cdot \mathbf{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3=|\mathbf{u}||\mathbf{v}|\cos \theta ,$ jahaan $\theta$ unke beech ka angle hai aur $\cos \theta$ ("cosine") measure karta hai ki woh kitne aligned hain: $\cos 0^{\circ }=1$ (same direction), $\cos 90^{\circ }=0$ (perpendicular), $\cos 180^{\circ }=-1$ (opposite).

Do formulas agree kyun karte hain? (outline) $\mathbf{u}$ aur $\mathbf{v}$ ko tail-to-tail rakho. Unke tips ko join karne wali straight line vector $\mathbf{u}-\mathbf{v}$ hai, aur uski length triangle geometry ki cosine rule ko obey karti hai: $|\mathbf{u}-\mathbf{v}{|}^2=|\mathbf{u}{|}^2+|\mathbf{v}{|}^2-2|\mathbf{u}||\mathbf{v}|\cos \theta$. Ab left side ko component rule se expand karo (slot by slot $(\mathbf{u}-\mathbf{v})\cdot (\mathbf{u}-\mathbf{v})$ multiply out karo): sab kuch cancel ho jaata hai sivaay $-2(u_1{v}_1+u_2{v}_2+u_3{v}_3)$ ke. Dono expressions ko match karne par $u_1{v}_1+u_2{v}_2+u_3{v}_3=|\mathbf{u}||\mathbf{v}|\cos \theta$ force hota hai. Toh "multiply-and-add" recipe aur "angle" meaning same number hain do alag costumes mein.

Yeh particular tool kyun? Topic constantly poochta hai "kya yeh arrow us arrow ke perpendicular hai?" Dot product ek single test se jawab deta hai: dot product exactly tab zero hota hai jab do vectors right angles par hote hain (kyunki $\cos 90^{\circ }=0$). Yahi plane ke normal form $(\mathbf{r}-\mathbf{a})\cdot \mathbf{n}=0$ ka poora engine hai. (Is page ke end mein VERIFY block neeche diye perpendicular example ko real numbers par check karta hai.)

Figure s03 do perpendicular arrows $\mathbf{u}$ aur $\mathbf{v}$ ko little right-angle square ke saath draw karta hai; unka dot product $0$ hai.

Figure s03 — zero dot product exactly do arrows ke beech right angle hai.

Full details Vectors and scalar (dot) product mein hain aur iska geometric use Vector projection mein.

Recall Perpendicular test

$(1,2,2)\cdot (2,-2,1)=?$ aur kya yeh perpendicular hain? ::: $2-4+2=0$, toh haan, perpendicular.

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7. Cross product — $\mathbf{u}\times \mathbf{v}$

Cross product do vectors leta hai aur ek naya vector return karta hai jo dono ke perpendicular hota hai ek saath. Iske components: $\mathbf{u}\times \mathbf{v}=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1).$

Yeh formula kuch perpendicular kyun deta hai? (outline) Perpendicular matlab "dot product zero" (§6). Toh test karo: $\mathbf{u}\times \mathbf{v}$ ko $\mathbf{u}$ ke saath dot karo aur sab multiply out karo — six terms pairs mein cancel ho jaate hain aur aapko exactly $0$ milta hai; same hota hai $\mathbf{v}$ ke saath dot karne par. (Is page ke end mein VERIFY block exactly yahi real numbers par check karta hai.) Woh built-in cancellation isliye hai ki topic exactly yahi component recipe use karta hai. Kaun sa direction point karta hai do opposite perpendicular directions mein se yeh §0 ke right-handed axes se fix hota hai.

Parallel edge-case (isliye ek plane ko non-parallel spanning arrows chahiye). $\mathbf{u}\times \mathbf{v}$ ki length $|\mathbf{u}||\mathbf{v}|\sin \theta$ ke barabar hai. Agar $\mathbf{u}$ aur $\mathbf{v}$ parallel hain (ya ek zero hai), toh $\theta =0^{\circ }$ (ya $180^{\circ }$), toh $\sin \theta =0$ aur cross product zero vector $0$ mein collapse ho jaata hai — jo kahin point nahi karta aur normal ke roop mein useless hai. Exactly isliye parent insist karta hai ki plane ke do spanning directions $\mathbf{u},\mathbf{v}$ non-parallel hone chahiye: tabhi $\mathbf{u}\times \mathbf{v}$ ek genuine, nonzero normal produce karta hai.

Yeh particular tool kyun? Ek plane do in-plane directions $\mathbf{u},\mathbf{v}$ se span hoti hai. Iska normal dono ke perpendicular bahar nikalna chahiye. Koi single dot product yeh nahi de sakta — lekin cross product designed hai do inputs ke orthogonal ek arrow output karne ke liye. Isliye parent likhta hai $\mathbf{n}=\mathbf{u}\times \mathbf{v}$.

Figure s04 ek shaded plane mein do arrows aur red cross product ko seedha uski taraf se bahar nikalta hua dikhata hai.

Figure s04 — cross product (red) dono arrows ke perpendicular khada hai; yeh zero ho jaata hai agar woh parallel hain.

Zyada Cross product mein.

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8. Normal vector — $\mathbf{n}$

Ek plane ka normal $\mathbf{n}$ koi bhi arrow hai jo plane se seedha bahar point karta hai, uske andar lie karne wali har direction ke right angles par. Yeh plane ke along nahi hai — yeh plane ke across hai. Yeh ek line ke direction $\mathbf{d}$ se opposite role hai, jo line ke along lie karta hai.

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Prerequisite map

Neeche diagram ek dependency chart hai: ise top se bottom padhein. Har box is note se ek idea hai, aur arrow "$A\to B$" matlab "aapko $A$ chahiye pehle $B$ samajhne se." Coordinates vectors banate hain; vectors add/subtract/scale/measure hote hain; woh do special products ko feed karte hain; aur sab kuch parent topic mein funnel hota hai. (Agar diagram aapke reader mein render nahi hota, same order exactly is page ka §0 → §8 hai.)

Sab kuch parent topic par converge hota hai.

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Equipment checklist

Answers cover karo aur khud test karo — agar koi ek bhi shaky hai, parent note se pehle us section ko dobara padho.

• Ek vector teen numbers store karta hai matlab… ::: $x,y,z$ axes mein se har ek ke along kitna step karna hai.
• Ek point aur position vector mein farq hai… ::: point ek fixed spot hai; position vector woh arrow hai origin $O$ se jo us spot tak reach karta hai (same teen numbers).
• $\mathbf{r}-\mathbf{a}$ woh arrow hai jo… ::: anchor point $A$ se roaming point $R$ tak jaata hai (difference arrow).
• $t\mathbf{d}$ geometrically arrow $\mathbf{d}$ ke saath kya karta hai… ::: ise factor $t$ se stretch/shrink karta hai (aur flip karta hai agar $t<0$), direction rakhe hue.
• "Do vectors parallel hain" symbols mein matlab… ::: ek doosre ka scalar multiple ke barabar hai, $\mathbf{u}=t\mathbf{v}$.
• Kya zero vector ek line ke liye valid direction hai? ::: Nahi — iska koi direction nahi hai; line ka $\mathbf{d}$ satisfy karna chahiye $\mathbf{d}\ne 0$.
• Parameter $t\in \mathbb{R}$ ek line mein kya contribute karta hai… ::: $t$ ki har value ek point pick karti hai; saari values poori line sweep karti hain.
• $|\mathbf{v}|$ compute hota hai… ::: $\sqrt{v_1^2+v_2^2+v_3^2}$ se, Pythagoras se arrow ki length.
• Unit vector $\hat{\mathbf{n}}$ hai… ::: $\mathbf{n}$ apni length $|\mathbf{n}|$ se divided (requires $\mathbf{n}\ne 0$), length exactly 1 deta hai.
• Dot product $\mathbf{u}\cdot \mathbf{v}$ exactly tab zero hota hai jab… ::: do vectors perpendicular hain ($90^{\circ }$).
• Cross product $\mathbf{u}\times \mathbf{v}$ ek vector output karta hai jo… ::: dono $\mathbf{u}$ aur $\mathbf{v}$ ke perpendicular hai (side right-hand rule se fix hoti hai).
• $\mathbf{u}\times \mathbf{v}$ kya hota hai jab $\mathbf{u}$ aur $\mathbf{v}$ parallel hain? ::: zero vector $0$ (kyunki $\sin \theta =0$) — isliye plane spanning directions non-parallel hone chahiye.
• Ek plane ka normal $\mathbf{n}$ point karta hai… ::: seedha plane se bahar (across), ek line ke direction se alag jo uske along lie karta hai.

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Connections

• Vectors and scalar (dot) product — yahan banaya gaya dot product plane ke normal form ko power karta hai.
• Cross product — woh perpendicular-to-both tool jo normals banata hai.
• Vector projection — distances ke liye dot product aur unit vectors use karta hai.
• Parametrisation and parameters — dials $t$ aur $s$ ka matlab.
• Systems of linear equations — jahaan yeh symbols saath solve hote hain.
• Intersection of lines and planes — payoff jab symbols fluent ho jaate hain.

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