Foundations — Vector representation — magnitude, direction, components

1.1.7 · D1 · Physics › Measurement, Vectors & Kinematics › Vector representation — magnitude, direction, components

Is page par assume kiya gaya hai ki aap kuch nahi jaante. Parent note padhne se pehle, aapko kuch giroh symbols aur pictures pehchanne ki zaroorat hai. Hum unhe ek ek karke banate hain, aur har ek sirf unhi cheezon ka use karta hai jo usse upar hain. Agar aapne kabhi koi arrow number ki tarah use hote nahi dekha, to yahan line ek se shuru karein.

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1. Arrow khud — ek vector kaisa dikhta hai

Formulas bhool jaao. Ek vector ek bana hua arrow hai: iske paas ek starting tail, ek pointed head, ek length, aur ek slant hota hai.

• Plain-words meaning: "itna, is direction mein."
• Picture: upar ka orange arrow. Uski length "itna" hai; jis taraf woh jhuka hai woh "is direction" hai.
• Topic ko iske zaroorat kyun hai: forces, velocities aur displacements sab ek specific amount ek specific taraf push karte hain. Arrow hi ek aisa object hai jo dono ko ek saath store karta hai.

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2. Playground — axes, origin, +x aur +y

Ek picture ko numbers mein badle ke liye hame ek ruler grid chahiye jis par arrow draw kar sakein.

• Plain-words meaning: graph paper jisme ek marked centre aur do marked "positive" directions hain — daayein ($+x$) aur upar ($+y$).
• Picture: upar wala navy cross; magenta dot origin hai; orange arrow labeled $+x$ daayein point karta hai, violet arrow labeled $+y$ upar point karta hai.
• Topic ko iske zaroorat kyun hai: hum hamesha ek vector origin se start karke draw karte hain. Tab "kitna daayein" (along $+x$) aur "kitna upar" (along $+y$) grid se simple readings ban jaati hain. Bina fixed grid ke, "kis taraf" ka koi matlab nahi.

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3. Angle $\theta$ — "kis taraf" ek number ke roop mein

• Plain-words meaning: aapne "daayein point karne" se kitna rotate kiya hai.
• Picture: upar wala violet wedge, $+x$ arrow se vector tak khulta hua.
• Topic ko iske zaroorat kyun hai: direction vector ki khabar ka aadha hissa hai. $\theta$ woh aadha hissa hai, ek number ke roop mein likha gaya taaki hum usse compute kar sakein.

To vector ki do-number description hai $(A,\theta )$ — length aur angle. Ise polar form kehte hain.

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4. Right triangle — arrow ke andar chupi machine

Ab key move. Arrow ke head se seedha neeche x-axis tak ek line daalo. Dekho kya appear hota hai:

• Picture: orange arrow hypotenuse hai; horizontal navy leg x-axis ke saath chalti hai; vertical magenta leg neeche daali gayi line hai.
• Topic ko iske zaroorat kyun hai: yahi triangle hai jo hame dono descriptions ke beech swap karne deta hai. Hypotenuse arrow ki length $A$ hai; corner angle $\theta$ hai; do legs woh numbers hain jo hum next section mein chahte hain.

Notice karein ki horizontal leg jawaab deti hai "arrow kitna daayein gaya?" aur vertical leg jawaab deti hai "kitna upar?" Woh do jawab components hain.

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5. Trig ratios — ek slant ko ratio mein badalna

Component ke kisi bhi formula likhne se pehle hame woh tools banana padenge jo angle $\theta$ ko triangle ki sides se connect karte hain. Woh tools trigonometric ratios hain. Yeh section component formulas se pehle jaanbujhkar aaya hai — aap $\cos$ aur $\sin$ tab tak use nahi kar sakte jab tak unka koi matlab na ho.

• ${\tan }^{-1}$ kyun matter karta hai: do legs se wapas ek angle par jaate hue, hmare paas sirf unka ratio hota hai. ${\tan }^{-1}$ woh tool hai jo ratio se angle padhta hai.

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6. Components — "kitna daayein / kitna upar" numbers

• Picture: triangle figure mein (Section 4), $A_x$ navy horizontal leg ki length hai, $A_y$ magenta vertical leg ki length hai.
• Topic ko iske zaroorat kyun hai: slanted arrows add karna mushkil hai. Horizontal aur vertical pieces ruler par ordinary numbers ki tarah add ho jaate hain. Components arrow ko do aasaan movements mein rewrite karte hain ("daayein jao, phir upar jao").
• Signs: agar arrow baayein reach karta hai, to $A_x$ negative hai; agar neeche reach karta hai, to $A_y$ negative hai. Sign ek direction flag hai.

Ab kyunki $\cos$ aur $\sin$ exist karte hain (Section 5), hum bridge likh sakte hain. Triangle se $\cos \theta =A_x/A$ aur $\sin \theta =A_y/A$ lo, phir dono sides ko hypotenuse $A$ se multiply karo:

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7. Har quadrant aur har degenerate case

Ek hi pair $A_x=A\cos \theta ,A_y=A\sin \theta$ pehle se har direction handle karta hai, flat aur straight-up cases bhi. Dekho ki signs kaise flip hote hain jab arrow rotate karta hai.

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8. Pythagoras — length recover karna

• Plain-words meaning: straight-line length "daayein jao, upar jao" rectangle ke diagonal ke barabar hoti hai.
• Picture: wahi right triangle — do legs aur hypotenuse.
• Topic ko iske zaroorat kyun hai: yeh do components se magnitude $A$ recover karta hai. Yeh "East phir North → straight-line distance" rule hai. Note karo yeh degenerate cases mein bhi kaam karta hai: agar $A_y=0$ to $A=\sqrt{A_x^2}=|A_x|$, jaisa expected hai.

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9. Unit vectors $\hat{i},\hat{j}$ — chhote signposts

Pehle hame exactly batana hoga ki chhoti hat ka matlab kya hai, kyunki ek newcomer ise guess nahi kar sakta.

• Plain-words meaning: ek "ek step East" arrow aur ek "ek step North" arrow.
• Hat ka matlab length one kyun hai: kyunki precisely yahi hat operator karta hai — ek arrow ko uski apni length se divide karo aur tumhare paas length-one copy reh jaati hai. (Aap Pythagoras se check kar sakte hain: components $(0.6,0.8)$ wale arrow ki length $\sqrt{0.6^2+0.8^2}=1$ hoti hai, to use hat milta hai.)
• Topic ko iske zaroorat kyun hai: component form $\vec{A}=A_x\hat{i}+A_y\hat{j}$ likhne ke liye, jo literally padhta hai "$A_x$ steps East jao, phir $A_y$ steps North jao." $\hat{i},\hat{j}$ har number ko batate hain ki woh kis direction mein apply hota hai.

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10. Poora symbol dictionary

Recall Har symbol ek card par (click to reveal)

$\vec{A}$ ::: ek vector — size aur direction wala arrow. $A$ ya $|\vec{A}|$ ::: magnitude — arrow ki length, hamesha $\ge 0$. $\theta$ ::: direction angle, $+x$ se anticlockwise. $+x,+y$ ::: positive grid directions — daayein aur upar. $A_x$ ::: x-component — kitna daayein (adjacent leg); negative agar arrow baayein point kare. $A_y$ ::: y-component — kitna upar (opposite leg); negative agar arrow neeche point kare. $\cos \theta$ ::: adjacent ÷ hypotenuse. $\sin \theta$ ::: opposite ÷ hypotenuse. $A_x=A\cos \theta$ ::: polar form se x-component (x-side adjacent hai → cosine). $A_y=A\sin \theta$ ::: polar form se y-component (y-side opposite hai → sine). $\tan \theta$ ::: opposite ÷ adjacent = steepness. ${\tan }^{-1}$ ::: tan ka undo — "is ratio wala angle kaunsa hai?"; undefined jab $A_x=0$.  ​ ::: square root — squaring undo karta hai; $A=\sqrt{A_x^2+A_y^2}$ mein use hota hai. $\hat{}$ ::: hat operator — "length 1 tak shrink karo"; unit vector mark karta hai. $\hat{i},\hat{j}$ ::: $+x$ aur $+y$ ke saath length-1 signpost arrows.

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Prerequisite map

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Equipment checklist

Khud test karo — tum parent note ke liye ready ho tabhi jab tum bina dekhay har sawaal ka jawaab de sako.

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Connections

• Scalars vs Vectors — kyun ek number size aur direction dono nahi rakh sakta.
• Unit Vectors i, j, k — yahan bane signposts $\hat{i},\hat{j}$, 3D tak extend kiye gaye.
• Vector representation — magnitude, direction, components — parent topic jisme ye foundations jaati hain.
• Vector Addition — Triangle & Parallelogram Law — jahan components kaam aate hain.
• Resolution of Forces — forces par apply kiye gaye trig ratios.
• Dot Product and Cross Product — components par built.
• Projectile Motion — velocity ko $v_x,v_y$ mein split karna.