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

1.1.10 · Physics › Measurement, Vectors & Kinematics

Unit vector KYA hota hai?

YEH KYUN chahiye? Kyunki yeh humein kisi bhi vector ko numbers ke simple collection ki tarah likhne dete hain: $\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$ Yahan $A_x,A_y,A_z$ components hain — vector har axis ke saath kitna reach karta hai. Hats direction carry karte hain; numbers amount carry karte hain.

Magnitude KAISE nikaalein (scratch se derive karein)

Derivation (pehle 2D mein): $xy$-plane mein $A_x$ aur $A_y$ ek right triangle ki do perpendicular sides hain jiska hypotenuse $\vec{A}$ hai. Toh $|\vec{A}{|}^2=A_x^2+A_y^2.$

3D mein extend karo: Base plane mein diagonal ko $r=\sqrt{A_x^2+A_y^2}$ ek side maano, aur $A_z$ (poore base ke perpendicular) ko doosri side. Phir Pythagoras: $|\vec{A}{|}^2=r^2+A_z^2=A_x^2+A_y^2+A_z^2.$

Unit vector KAISE construct karein

Derivation: Maano $\vec{A}$ ka magnitude $|\vec{A}|$ hai. Define karo $\hat{A}=\frac{\vec{A}}{|\vec{A}|}.$ Iska magnitude hai $|\hat{A}|=\frac{|\vec{A}|}{|\vec{A}|}=1.✓$ Ek vector ko positive scalar $1/|\vec{A}|$ se multiply karne par woh rotate nahi hota, toh direction preserved rehti hai. Ho gaya.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Vectors — components and resolution
• Magnitude and direction of a vector
• Vector addition — triangle and parallelogram laws
• Dot product (uses $\hat{i}\cdot \hat{i}=1,\hat{i}\cdot \hat{j}=0$)
• Cross product (uses $\hat{i}\times \hat{j}=\hat{k}$)
• Position vector and displacement
• Pythagoras theorem

Concept Map