Worked examples — Geometric interpretation — signed volume

4.5.23 · D3 · Maths › Linear Algebra (Full) › Geometric interpretation — signed volume

Yeh page har us case ka drill hai jo determinant-as-signed-volume idea tumhare saamne rakh sakta hai. Pehle hum saare scenarios ka ek map banate hain, phir har cell ke liye ek example karte hain taaki koi bhi situation unfamiliar na lage.

Yahan sab kuch parent idea pe tikaa hai: Geometric interpretation — signed volume. Agar koi symbol pehchaana na aaye, toh woh wahan earn kiya gaya tha — lekin main har ek ko aate waqt re-anchor karunga.

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Scenario matrix

Is topic ka har determinant problem inhi cells mein se ek hai. "Covered by" column ko ek promise ki tarah padho: har ek neeche work out kiya gaya hai.

#	Case class	Isme kya khaas hai	Covered by
A	2D, dono signs positive area	ordinary slanted box, orientation rakhi gayi	Ex 1
B	2D, negative determinant	ek reflection/flip, sign $-$ hai	Ex 2
C	2D, zero determinant	edges parallel → box ek line mein squash ho gaya	Ex 3
D	2D, near-degenerate (limiting)	edges almost parallel → area $\to 0$	Ex 4
E	3D, right-handed frame	positive triple product	Ex 5
F	3D, left-handed frame	ek axis flip → negative	Ex 6
G	Poori matrix ko scale karna	$\det (cA)=c^n\det A$, woh $n$-th power trap	Ex 7
H	Real-world word problem	ek plot ka area / physical volume, sign discard karo	Ex 8
I	Exam twist (row op invariance)	ek column ko shear karne se volume nahi badlta	Ex 9
J	Non-square "volume"	Gram determinant $\sqrt{\det (A^{\top }A)}$ chahiye	Ex 10

Prerequisite links Determinant — definition and cofactor expansion, Cross product and scalar triple product, Linear independence and rank aur Change of variables and the Jacobian mein hain.

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Worked examples

Ex 1 — Cell A: ek ordinary slanted box

Step 1. Vectors ko columns ki tarah rakho aur 2D formula $\det \left(\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right)=a_1{b}_2-a_2{b}_1$ use karo.

Yeh step kyun? Parent derivation ne prove kiya ki yeh cross-multiplication hi signed area hai — koi aur cheez multilinear/alternating/normalized axioms ko satisfy nahi karti.

Step 2. Values daalo: $a_1=3,a_2=1,b_1=1,b_2=2$. $\det =(3)(2)-(1)(1)=6-1=5.$

Yeh step kyun? $a_1{b}_2$ box ki "spread" hai achhe rotational direction mein; $a_2{b}_1$ us part ko subtract karta hai jo shear ko double-count kar leta.

Verify: Figure dekho — laal parallelogram clearly ek $3\times 3$ box (area 9) ke andar fit hota hai aur uska aadhe se zyada hissa hai, toh 5 theek lagta hai. Sign $+$ hai kyunki $\mathbf{a}\to \mathbf{b}$ counterclockwise hai. ✓

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Ex 2 — Cell B: ek reflection (negative determinant)

Step 1. Pehle "natural" order $\det [\mathbf{a}\mathbf{b}]=(2)(2)-(1)(1)=3$ lo.

Yeh step kyun? Magnitude ($3$) ko anchor karna se hum isolate kar sakte hain ki swap akela kya karta hai.

Step 2. Ab swapped order $\det [\mathbf{b}\mathbf{a}]=(1)(1)-(2)(2)=1-4=-3$.

Yeh step kyun? Alternating axiom kehta hai ki do edges swap karne se sign flip hota hai: yeh wahi eit hai peeche se dekhi, toh counterclockwise clockwise ban gaya hai.

Verify: $|-3|=3=|+3|$ — size unchanged hai, sirf SIDE flip hui, bilkul waise jaisa Orientation and handedness of bases predict karta hai. Negative ka matlab "negative area" nahi hai, yeh ek mirrored orientation hai. ✓

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Ex 3 — Cell C: ek squashed box (zero)

Step 1. $\det =(2)(6)-(4)(3)=12-12=0.$

Yeh step kyun? Jab ek edge doosri ka scalar multiple ho, box ek segment mein collapse ho jaata hai — koi area enclose karne ko nahi.

Step 2. Interpret karo: zero determinant $⟺$ columns linearly dependent hain $⟺$ map non-invertible hai.

Yeh step kyun? Yeh Linear independence and rank aur Invertibility and the inverse matrix ka bridge hai — ek flat box un-flatten nahi ho sakta.

Verify: $\mathbf{b}-\frac{3}{2}\mathbf{a}=(0,0)$, exact parallelism confirm karta hai, toh area $0$ hi hona chahiye. ✓

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Ex 4 — Cell D: limiting case (almost parallel)

Step 1. $\det =(1)(\epsilon )-(0)(1)=\epsilon .$

Yeh step kyun? Formula shrinking ko explicit banata hai — area hi woh tiny height $\epsilon$ hai.

Step 2. Limit lo: $\underset{\epsilon \to 0}{\lim }\epsilon =0$.

Yeh step kyun? Hum ek limit use karte hain (sirf $\epsilon =0$ plug karne ki bajay) approach dekhne ke liye: box flat pe jump nahi karta, continuously deflate hota hai. Yahi continuity reason hai ki determinants Change of variables and the Jacobian mein achhe se behave karte hain.

Verify: $\epsilon =0.1$ par, area $=0.1$; $\epsilon$ ko halve karo toh area halve hoti hai — linear collapse, koi surprise nahi. $\epsilon =0$ par hum exactly Cell C pe pahunch jaate hain. ✓

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Ex 5 — Cell E: ek right-handed 3D frame

Step 1. Triple product $\det [\mathbf{a}\mathbf{b}\mathbf{c}]=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$ use karo.

Yeh step kyun? $\mathbf{b}\times \mathbf{c}$ (cross product, dekho Cross product and scalar triple product) ek vector hai jiska length base-parallelogram ka area hai aur jo us base se perpendicular point karta hai. $\mathbf{a}$ ke saath dot karna base-area ko height-along-that-normal se multiply karta hai = volume.

Step 2. $\mathbf{b}\times \mathbf{c}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 1& 0\\ 1& 1& 1\end{array}\right|=\mathbf{i}(1\cdot 1-0\cdot 1)-\mathbf{j}(1\cdot 1-0\cdot 1)+\mathbf{k}(1\cdot 1-1\cdot 1)=(1,-1,0).$

Yeh step kyun? Symbol-grid ka determinant sirf cross product ke components ka formula hai.

Step 3. $\mathbf{a}\cdot (1,-1,0)=(1)(1)+(0)(-1)+(0)(0)=1.$

Yeh step kyun? Dot product height component pick karta hai; yahan woh exactly $1$ hai.

Verify: Shears volume preserve karte hain (woh layers ko sideways slide karte hain bina squeeze kiye), toh ek sheared unit cube ka volume $1$ hona chahiye. Sign $+$ → right-handed. ✓

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Ex 6 — Cell F: ek axis flip karke left-handed banana

Step 1. $\mathbf{b}\times {\mathbf{c}}^{\prime }=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 1& 0\\ 1& 1& -1\end{array}\right|=\mathbf{i}(1\cdot (-1)-0)-\mathbf{j}(1\cdot (-1)-0)+\mathbf{k}(0)=(-1,1,0).$

Yeh step kyun? Same recipe; teesre row ka sign flip hone se normal ki direction flip ho gayi.

Step 2. $\mathbf{a}\cdot (-1,1,0)=(1)(-1)+0+0=-1.$

Yeh step kyun? Ab $\mathbf{a}$ base plane ke negative side par baitha hai → negative height → negative signed volume.

Verify: $|-1|=1$, Ex 5 ki size se match karta hai; sirf SIDE badli. Yeh exactly ek handedness reversal hai — right hand left hand ban gaya. ✓

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Ex 7 — Cell G: poori matrix ko scale karna

Step 1. $5A=\left(\begin{array}{cc}10& 5\\ 0& 15\end{array}\right)$, toh $\det (5A)=(10)(15)-(5)(0)=150.$

Yeh step kyun? Directly compute karne se trap expose hota hai shortcut trust karne se pehle.

Step 2. Rule $\det (cA)=c^n\det A$ se $n=2$ ke saath match karo: $5^2\cdot 6=25\cdot 6=150.$

Yeh step kyun? Har edge ko $c$ se scale karna ek $n$-dimensional volume ko $c$ se per dimension scale karta hai — wahi $n$-th power hai.

Verify: $150=5^2\times 6$. ✓ 3D mein same matrix idea $c^3$ deta; exponent hamesha dimension hota hai.

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Ex 8 — Cell H: ek real-world plot of land

Step 1. Signed area $=\det =(40)(50)-(10)(15)=2000-150=1850.$

Yeh step kyun? Determinant signed area deta hai; zameen ke liye hum phir magnitude lete hain.

Step 2. Physical area $=|1850|=1850{\text{m}}^2$.

Yeh step kyun? Orientation ka ek plot ke liye koi matlab nahi — hum sirf SIZE rakhte hain. Sign discard karna sahi real-world move hai (parent note mein $\det =-1$ wali galti sign ko "impossible" padhne ke baare mein warn karti hai).

Verify: Bounding box $40\times 50=2000{\text{m}}^2$; parallelogram thoda kam hai, aur $1850<2000$ sahi order of magnitude ke saath. Units $\text{m}\times \text{m}={\text{m}}^2$ ✓.

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Ex 9 — Cell I: exam twist (ek column ka multiple add karna)

Step 1. Multilinearity se, $\det [\mathbf{a}{\mathbf{b}}^{\prime }]=\det [\mathbf{a}\mathbf{b}]+4\det [\mathbf{a}\mathbf{a}].$

Yeh step kyun? Multilinearity ek column ke andar sum ko do determinants mein split karne deti hai.

Step 2. Alternating axiom doosri term ko khatam karta hai: $\det [\mathbf{a}\mathbf{a}]=0$. Toh $\det [\mathbf{a}{\mathbf{b}}^{\prime }]=\det [\mathbf{a}\mathbf{b}]=5.$

Yeh step kyun? Do equal edges wala box flat hota hai → $0$; yahi poora reason hai ki column-shear operations determinant preserve karti hain.

Step 3. Numeric check: ${\mathbf{b}}^{\prime }=(1,2)+4(3,1)=(13,6)$. Phir $\det =(3)(6)-(1)(13)=18-13=5.$

Yeh step kyun? Axiom argument ko brute force se confirm karta hai.

Verify: $5=5$. ✓ Column operations ke under yeh invariance hi reason hai ki Gaussian elimination $\det$ preserve karta hai (row swaps tak) — Invertibility and the inverse matrix ke liye ek load-bearing fact.

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Ex 10 — Cell J: non-square, Gram determinant use karo

Step 1. $M=[\mathbf{a}\mathbf{b}]$ (size $3\times 2$) banao. Sahi area hai Gram determinant $\sqrt{\det (M^{\top }M)}$.

Yeh step kyun? Non-square $M$ ke liye $\det M$ undefined hai. $M^{\top }M$ columns ke beech dot products ka woh chhota $2\times 2$ matrix hai; uska determinant squared area recover karta hai ambient dimension se independent ho kar.

Step 2. $M^{\top }M=\left(\begin{array}{cc}\mathbf{a}\cdot \mathbf{a}& \mathbf{a}\cdot \mathbf{b}\\ \mathbf{b}\cdot \mathbf{a}& \mathbf{b}\cdot \mathbf{b}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 25\end{array}\right).$

Yeh step kyun? $\mathbf{a}\cdot \mathbf{a}=1$, $\mathbf{b}\cdot \mathbf{b}=0+9+16=25$, aur $\mathbf{a}\cdot \mathbf{b}=0$ (perpendicular hain).

Step 3. $\det (M^{\top }M)=(1)(25)-0=25$, toh area $=\sqrt{25}=5.$

Yeh step kyun? Square root "squared area" ko wapas area mein convert karta hai — $|\det |$ ka extension un shapes ke liye jo space se lower dimension ki hain.

Verify: Lengths $1$ aur $5$ ke perpendicular vectors ek rectangle span karte hain jiska area $5$ hai. ✓ Forecast se exactly match karta hai.

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Active recall

Recall Answers cover karo

Do columns swap karne se $|\det |$ badlta hai kya? ::: Nahi — sirf sign flip hoti hai; magnitude unchanged rahti hai. Ek $2\times 2$ matrix ke liye $\det A=6$; $\det (5A)$ kya hai? ::: $5^2\cdot 6=150$. Do edges parallel hain. Signed area kya hai? ::: Exactly $0$ (flattened box). 3D mein rehne wale do vectors ka area kaise nikaalte hain? ::: $\sqrt{\det (M^{\top }M)}$, Gram determinant. Column $\mathbf{b}$ mein $4\mathbf{a}$ add karne se $\det$ par kya asar hota hai? ::: Kuch nahi — ek shear determinant preserve karti hai.

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Connections

• Geometric interpretation — signed volume (parent)
• Determinant — definition and cofactor expansion
• Cross product and scalar triple product
• Linear independence and rank
• Invertibility and the inverse matrix
• Change of variables and the Jacobian
• Eigenvalues — product equals determinant
• Orientation and handedness of bases