Vector representation — magnitude, direction, components: Question bank

1.1.7 · D5 · Physics › Measurement, Vectors & Kinematics › Vector representation — magnitude, direction, components

Definition Is poore page par use hone wali notation

Kisi bhi trap se pehle, symbols lock in karo taaki koi line surprise na kare:

$A$ — arrow khud (bold-with-arrow ka matlab hai "yeh ek vector hai, iska ek direction hai").
$A = ∣ A ∣$ — magnitude, arrow ki seedhi length. Hamesha $A \geq 0$ (ek length negative nahi ho sakti).
$A_{x}, A_{y}$ — components: arrow horizontal (x) aur vertical (y) axis ke along kitna pahunchta hai. Yeh negative ho sakte hain (left/down point karte hue).
$\hat{i}, \hat{j}$ — unit vectors jinka length 1 hai $+ x, + y$ ke along; hum likhte hain $A = A_{x} \hat{i} + A_{y} \hat{j}$ .
$θ$ — direction angle, anticlockwise (CCW) +x axis se measure kiya gaya. Yeh ek convention hi har cos/sin aur quadrant rule ka adhar hai — x-side $θ$ ke adjacent hai, isliye use cosine milta hai.
$\hat{A}$ — "hat" ka matlab hai ek unit vector: $A$ jaise hi direction, lekin length exactly 1, jo bana hai $\hat{A} = A / A$ se.

Figure — Vector representation — magnitude, direction, components

True or false — justify

True or false: Ek vector ki magnitude negative ho sakti hai agar woh down-left point kare.

False — magnitude hai

A = A_{x}^{2} + A_{y}^{2} \geq 0

hamesha; "down-left" ki saari khabar sirf components ke signs aur angle mein hoti hai, size mein kabhi nahi.

True or false: Do vectors jinka magnitude same ho woh same vector hote hain.

False — ek vector ko dono magnitude aur direction chahiye; do arrows equally lambe ho sakte hain phir bhi alag direction mein point kar sakte hain, isliye woh alag vectors hain.

True or false: X-component hamesha cosine use karta hai, chahe angle kaise bhi draw kiya ho.

False — cosine us axis ke saath jaata hai jis axis se angle measure kiya gaya ho. Yahaan

θ

+x se CCW measure hai, isliye x-side adjacent hai aur cosine milta hai; agar

θ

+y axis se measure hota, toh y-side adjacent hoti.

True or false:

∣ A + B ∣ = ∣ A ∣ + ∣ B ∣

kisi bhi do vectors ke liye.

False — arrows partly cancel ho sakte hain, isliye resultant usually chota hota hai sum se; equality tab hi hoti hai jab

A

aur

B

same direction mein hon (parallel).

True or false: Unit vector ki magnitude 1 hoti hai lekin koi direction nahi hota.

False — unit vector

\hat{A}

pure direction hai; woh original ki direction rakhta hai aur sirf size hatata hai, isliye uski magnitude exactly 1 hoti hai.

True or false: Agar kisi vector ka

A_{x} = 0

hai, toh uska angle undefined hai.

False — woh seedha

\pm y

ke along point karta hai, isliye

θ = 90°

(agar

A_{y} > 0

) ya

270°

(agar

A_{y} < 0

); zero se divide karke formula

tan^{- 1} (A_{y} / A_{x})

toot jaata hai, direction khud nahi.

True or false: Ek vector ko

- 1

se multiply karne par uski magnitude unchanged rehti hai.

True — har component ka sign flip hota hai, lekin

(- A_{x})^{2} + (- A_{y})^{2} = A_{x}^{2} + A_{y}^{2}

; sirf direction reverse hoti hai (

180°

rotate hoti hai).

True or false:

30°

wala vector aur

390°

wala vector alag vectors hain.

False —

390° = 30° + 360°

, ek poora chakkar laga ke same direction par wapas, isliye (equal magnitude ke saath) woh identical arrows hain.

Why questions

Why does the x-component use cosine and not sine (angle CCW from +x)?

Kyunki

θ

x-axis se measure kiya jaata hai, jisse x-side angle ko chhooti hai (adjacent), aur cosine = adjacent/hypotenuse.

Why must you check quadrants after

tan^{- 1} (A_{y} / A_{x})

Kyunki

A_{y} / A_{x}

aur

(- A_{y}) / (- A_{x})

identical ratio dete hain, isliye tan ek arrow aur uske exact opposite mein farq nahi kar sakta; sirf components ke signs batate hain ki dono mein se kaun sa sahi hai.

Why does

atan2 (A_{y}, A_{x})

solve the quadrant ambiguity automatically?

Kyunki yeh

A_{y}

aur

A_{x}

ko do alag inputs ki tarah leta hai instead of unhe pehle se ek ratio mein divide karne ke, toh yeh har sign alag se padh sakta hai aur angle ko sahi quadrant mein rakh sakta hai — single ratio jo sign information throw away karta hai, exactly woh yeh rakhta hai.

Why can't a single number (scalar) represent a velocity fully?

Velocity mein "kitna fast" aur "kis direction mein" dono hote hain; akela number sirf ek cheez rakhta hai, isliye direction kho jaata — isliye hume do cheezein rakhne wala arrow chahiye.

Why do components add like ordinary numbers but whole arrows don't?

Components fixed perpendicular axes ke along hote hain, isliye x sirf x ke saath aur y sirf y ke saath interact karta hai; tedhe arrows directions mix karte hain, isliye woh partly cancel ho sakte hain aur pehle resolve karna padta hai.

Why is dividing by

A

the right way to build a unit vector?

Length se divide karna arrow ko length 1 tak rescale karta hai jab ki parts ka ratio preserve rehta hai, jo pure direction deta hai size normalize karke.

Why does magnitude stay

\geq 0

even for a vector pointing into Quadrant III?

Formula har component ko square karta hai, square root lene se pehle saari sign information mita ke — isliye magnitude ek length hai, aur lengths kabhi negative nahi hoti.

Edge cases

Edge case: Zero vector

0

ka direction angle kya hai?

Undefined — uski koi length nahi aur woh kahin nahi point karta, isliye koi meaningful angle exist nahi karta (aur

tan^{- 1} (0/0)

indeterminate hai).

Edge case:

A = 5 \hat{i} + 0 \hat{j}

—

θ

kya hai, aur kya formula kaam karta hai?

θ = 0°

; formula

tan^{- 1} (0/5) = 0°

cleanly kaam karta hai aur koi quadrant fix nahi chahiye kyunki

A_{x} > 0

Edge case:

A = 0 \hat{i} - 4 \hat{j}

—

θ

kya hai?

Yeh seedha neeche point karta hai, isliye

θ = 270°

(ya

- 90°

); formula

tan^{- 1} (- 4/0)

zero se divide karta hai, isliye direction geometrically padhna padta hai.

Edge case:

A = - 2 \hat{i} + 0 \hat{j}

— naive

tan^{- 1}

kya deta hai aur sach kya hai?

tan^{- 1} (0/ - 2) = 0°

, lekin arrow

- x

ke along point karta hai, isliye true angle

180°

hai;

A_{x} < 0

rule

180°

add karke fix karta hai.

Edge case: Exactly

θ = 90°

par ek vector — uske components kya hain?

A_{x} = A cos 90° = 0

aur

A_{y} = A sin 90° = A

; yeh poora +y ke along hai, jo confirm karta hai ki formulas boundary ko sahi handle karte hain.

Edge case: Kya zero vector ke liye unit vector exist kar sakta hai?

Nahi — tum

A = 0

se divide karoge, jo undefined hai; zero vector ka koi direction nahi hai jo normalize ho sake.

Edge case: Do vectors ki equal magnitude hai aur woh add karke zero dete hain — unka kya relationship hai?

Woh exact opposites hain (antiparallel), har ek doosre ka negative hai, isliye unke components component-by-component cancel hokar

0

dete hain.

Connections

Scalars vs Vectors — "ek number enough kyun nahi hai" wale traps.
Vector Addition — Triangle & Parallelogram Law — "magnitudes mat jodo" wala trap.
Resolution of Forces — jahaan cos/sin swap sabse zyada kaat ta hai.
Dot Product and Cross Product — sign aur direction ki subtleties aage bhi chalti hain.
Projectile Motion — velocity ke components action mein.
Unit Vectors i, j, k — pure-direction reasoning.

Question bank — Vector representation — magnitude, direction, components

True or false — justify

Spot the error

Why questions

Edge cases

Connections