Unit vectors — î, ĵ, k̂; constructing unit vector: Question bank

1.1.10 · D5 · Physics › Measurement, Vectors & Kinematics › Unit vectors — î, ĵ, k̂; constructing unit vector

Yeh parent Unit Vectors note ka companion hai. Koi naya formula introduce nahi ho raha — sirf woh tarikein hain jinse purane formulas galat use ho jaate hain.

True ya false — justify karo

Har vector jiska length 1 ho, unit vector hota hai.

True — yahi toh definition hai. "Unit" ka matlab hai magnitude exactly

1

; iska direction se koi lena-dena nahi hai.

\hat{i} + \hat{j}

ek unit vector hai.

False — iska magnitude

1^{2} + 1^{2} = 2 \approx 1.41

hai,

1

nahi. Do unit vectors ko add karne se unit vector nahi milta jab tak woh same vector na hon.

Ek unit vector ko kisi axis ke saath point karna zaroori hai.

False —

0.6 \hat{i} + 0.8 \hat{j}

ek bilkul sahi unit vector hai jo diagonally point karta hai. Sirf

\hat{i}, \hat{j}, \hat{k}

hi axes par hote hain; har direction ka apna unit vector hota hai.

Agar

\hat{A}

ek unit vector hai, toh

- \hat{A}

bhi ek unit vector hai.

True — negate karne se direction flip hoti hai lekin length

∣ - 1∣ = 1

se multiply hoti hai, toh magnitude

1

rehti hai. Yeh opposite direction mein point karta hai, same length ke saath.

Ek unit vector dimensionless hota hai, isliye use length se multiply karne par length milti hai.

True — unit vector sirf direction carry karta hai, koi units nahi, toh physical units poori tarah us scalar magnitude se aate hain jisse tum multiply karte ho (jaise metres, newtons).

Zero vector

0

ka ek unit vector hota hai.

False —

0

ka magnitude

0

hai, aur

\hat{A} = 0 /0

zero se divide karna hai. Jis vector ki koi length nahi, uski koi direction bhi nahi, toh koi unit vector exist nahi karta.

Do alag-alag vectors ek hi unit vector share kar sakte hain.

True —

3 \hat{i} + 4 \hat{j}

aur

6 \hat{i} + 8 \hat{j}

dono

\hat{A} = 0.6 \hat{i} + 0.8 \hat{j}

dete hain. Woh same direction mein point karte hain aur sirf length mein differ karte hain, jo unit vector discard kar deta hai.

\hat{i} \cdot \hat{i} = 1

lekin

\hat{i} \cdot \hat{j} = 0

True — har basis vector jab apne aap se dot product kare toh

∣ \hat{i} ∣∣ \hat{i} ∣ cos 0^{\circ} = 1

milta hai, aur perpendicular wale se dot karne par

cos 9 0^{\circ} = 0

milta hai. Yahi orthonormal property hai.

Error dhundo

A = 3 \hat{i} + 4 \hat{j}

, toh

∣ A ∣ = 3 + 4 = 7

Components perpendicular hote hain, isliye unhe Pythagoras se combine karte hain, addition se nahi:

3^{2} + 4^{2} = 5

. Simple addition sirf tab kaam karta jab vectors same line par hon.

A = 6 \hat{i} + 8 \hat{j}

ko unit vector mein shrink karne ke liye,

2

se divide karo aur

3 \hat{i} + 4 \hat{j}

lo."

Tumhe vector ki apni magnitude (

10

) se divide karna hoga, kisi arbitrary number se nahi.

2

se divide karne par length

5

milti hai,

1

nahi; sirf

A /∣ A ∣

hi length exactly

1

guarantee karta hai.

\hat{B} = \frac{1}{3} \hat{i} - \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k}

, lekin magnitude positive hai toh main

+ \frac{2}{3} \hat{j}

likhta hoon."

Minus sign magnitude ke baare mein nahi hai — yeh encode karta hai ki component

- y

mein point karta hai. Ise drop karne se direction badal jaati hai; sirf overall magnitude forced positive hoti hai, kabhi bhi individual signed components nahi.

∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}

, aur

A_{y} = - 2

, toh

A_{x}^{2} - 4 + A_{z}^{2}

milega."

Negative ko square karne par positive milta hai:

(- 2)^{2} = + 4

. Square sign erase kar deta hai, isliye magnitude hamesha non-negative hoti hai chahe component signs kuch bhi hon.

"Unit vector point karta hai, toh iske koi units nahi hote — matlab yeh actually ek vector nahi hai."

Yeh ek genuine vector hi hai (iska direction aur magnitude

1

hai); "no units" ka sirf matlab hai ki yeh dimensionless hai. Dimensionless aur "not a vector" ek cheez nahi hai.

A = ∣ A ∣ \cdot \hat{A}

, toh rebuild karne ke liye main magnitude aur unit vector add karta hoon."

Yeh multiplication hai, addition nahi: magnitude (ek scalar) times direction (unit vector). Ek scalar ko vector mein add karna defined operation hi nahi hai.

Kyunki teen basis vectors independent perpendicular axes ke saath point karte hain, toh unhe scale aur add karke space mein har point tak pahuncha ja sakta hai — hats direction supply karte hain, numbers amount supply karte hain, milke ek sum banate hain.

Component ka sign number mein kyun hota hai, hat mein kyun nahi?

Hats

+ x, + y, + z

ke saath fixed pointers hain; opposite direction mein point karne ke liye ek negative number lagaya jaata hai, jaise

- 2 \hat{j}

ka matlab hai "

- y

direction mein 2 units". Isse basis constant rehta hai.

Edge cases

Kisi vector ka unit vector kya hoga jo already magnitude

1

rakhta ho?

Vector khud hi —

1

se divide karne se kuch nahi badlta, toh

\hat{A} = A

. Yeh already ek pure pointer hai.

Sirf

\hat{i}

ka unit vector kya hai?

Sirf

\hat{i}

, kyunki

∣ \hat{i} ∣ = 1

. Standard basis vectors already unit vectors hain; kuch rescale nahi karna hai.

Kya ek unit vector exactly

\hat{i}

aur

\hat{j}

dono ho sakta hai ek saath?

Nahi — ek single unit vector ki ek direction hoti hai.

\hat{i}

aur

\hat{j}

alag-alag (perpendicular) axes ke saath point karte hain, toh yeh do distinct unit vectors hain, ek nahi.

"Construct a unit vector" ki recipe ka kya hoga agar ek component zero ho, jaise

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

Phir bhi kaam karta hai:

∣ A ∣ = 0 + 25 = 5

, toh

\hat{A} = \hat{j}

. Zero component ka matlab sirf yeh hai ki pointer ki us axis ke saath koi reach nahi hai — kuch break nahi hota.

Kya koi unit vector "koi direction nahi" mein point karta hai?

Nahi — woh zero vector hoga, jiska magnitude

0

hai, aur unit vector ki definition ke anusaar magnitude

1

chahiye. "Koi direction nahi" aur "unit length" contradictory hain.

Agar do vectors ki equal magnitude ho lekin opposite unit vectors hon, toh unka kya relation hai?

Woh ek doosre ke exact negatives hain: same length, directly opposite directions, toh

A = - B

(unke displacements cancel ho jaate hain).

Recall Yaad rakhne wale one-line takeaways

Unit = magnitude 1, not "chhota", not "dimensionless-isliye-vector-nahi".
Magnitude ke liye hamesha squares chahiye (Pythagoras), kyunki components perpendicular hain.
Signs numbers mein hote hain; hats fixed positive-axis pointers hain.
Zero vector woh ek vector hai jiska koi unit vector nahi hota.

Connections

Unit vectors — î, ĵ, k̂; constructing unit vector
Vectors — components and resolution
Magnitude and direction of a vector
Vector addition — triangle and parallelogram laws
Dot product
Position vector and displacement
Pythagoras theorem

Question bank — Unit vectors — î, ĵ, k̂; constructing unit vector

True ya false — justify karo

Error dhundo

Why questions

Edge cases

Connections