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

1.1.10 · D1 · Physics › Measurement, Vectors & Kinematics › Unit vectors — î, ĵ, k̂; constructing unit vector

Isse pehle ki tum parent note unit vectors padho, tumhe har woh symbol samajhna hoga jo woh quietly assume karta hai. Neeche, har ek cheez pichli cheez se build ki gayi hai: plain words → picture → yeh topic ise kyun chahta hai. Koi bhi cheez use nahi hoti jab tak woh draw na ho.

1. Arrow (vector) kya hota hai?

Ise ek ordinary number jaise $5$ (scalar) se compare karo — uske paas sirf size hoti hai, koi direction nahi. Temperature ek scalar hai; ek dhakka ek vector hai (tumhe pata hona chahiye kitni zyada taakat se AUR kis direction mein).

Yeh topic ise kyun chahta hai: "unit vector" ka poora idea tab hi samajh aata hai jab tum yeh maan lo ki ek arrow secretly do independent facts store karta hai. Figure mein red arrow dekho — uski length aur uska tilt do alag-alag dials hain.

2. Number line, axes, aur origin

Yeh topic ise kyun chahta hai: yeh kehne ke liye ki "yeh arrow $3$ right aur $4$ upar tak pahunchta hai" tumhe pehle ek right aur ek upar chahiye jiske against measure kar sako. Dekhiye Position vector and displacement ki points ko apne coordinates kaise milte hain.

3. Components — arrow ko "right" aur "up" amounts mein split karna

Figure dekho: arrow ki tip se $x$-axis par seedha neeche ek straight line daalo — jo shadow padta hai woh $A_x$ hai. $y$-axis par seedha across ek daalo — woh shadow $A_y$ hai. Arrow diagonal hai; uske do shadows components hain.

Signs matter karte hain. Agar arrow left ki taraf point karta hai, toh $A_x$ negative hoga. Agar yeh neeche ki taraf point karta hai, toh $A_y$ negative hoga. Component ka sign batata hai ki us axis ke saath do opposite directions mein se kaunsi hai. Ise kabhi mat chodo — parent note mein gire hue signs ko top mistake ke roop mein list kiya gaya hai. Poori kahani ke liye Vectors — components and resolution dekho.

4. Hat, aur î, ĵ, k̂ — pure pointers

Yeh topic inhe kyun chahta hai: yeh direction ki alphabet hain. $\vec{A}=A_x\hat{i}+A_y\hat{j}$ likhne ka matlab hai "$\hat{i}$ direction mein $A_x$ steps lo, phir $\hat{j}$ direction mein $A_y$ steps lo." Numbers amount carry karte hain; hats direction carry karte hain. Yahi clean split hai jis par yeh topic bana hua hai.

5. Arrow ko plain number se multiply karna (scaling)

Yeh topic ise kyun chahta hai: unit vector construct karna hi scaling hai ($\hat{A}=\vec{A}\cdot \frac{1}{|\vec{A}|}$), aur vector ko rebuild karna bhi scaling hai ($\vec{A}=|\vec{A}|\hat{A}$).

6. Magnitude aur square-root sign

Ise compute karne ke liye hume do symbols chahiye:

• Squaring, $A_x^2=A_x\times A_x$. Squaring hamesha kuch positive deta hai, isliye negative component apne positive twin ke barabar contribute karta hai — length ka koi sign nahi hota.
• Square root, $\sqrt{\cdot }$, squaring ko undo karna: $\sqrt{25}=5$ kyunki $5^2=25$.

Square root kyun, sirf components add kyun nahi? Kyunki components right triangle ki perpendicular sides hain — aur right triangle ki long side chhoti sides ka sum nahi hoti. Yeh Pythagoras theorem hai:

$\vec{A}=3\hat{i}+4\hat{j}$ ke liye: plain addition galat $3+4=7$ deta hai; Pythagoras sahi $\sqrt{3^2+4^2}=\sqrt{25}=5$ deta hai. Magnitude and direction of a vector dekho.

7. Degenerate case — zero vector

Yeh foundations topic ko kaise feed karte hain

Ise upar se neeche padho: arrows aur axes components dete hain; components ek right triangle banate hain; Pythagoras us triangle ko ek magnitude mein badalta hai; magnitude plus scaling plus basis hats tumhe unit vector dete hain.

Equipment checklist

Khud ko test karo — tum ready ho jab tum bina dekhey har line ka jawab de sako.

Connections

• Unit vectors — î, ĵ, k̂; constructing unit vector (parent note jiske liye yeh page tumhe prepare karta hai)
• Vectors — components and resolution
• Magnitude and direction of a vector
• Pythagoras theorem
• Position vector and displacement