Foundations — Position vector, displacement, distance

1.1.13 · D1 · Physics › Measurement, Vectors & Kinematics › Position vector, displacement, distance

Is page par koi assumption nahi hai. Har letter, hat, arrow, aur symbol jo parent note mein use hua hai, yahan ground up se build kiya gaya hai, ek aisi sequence mein jahan har idea sirf usse pehle wale ideas par depend karta hai.

0. Sabse pehli cheez: ek point aur ek dot jisse measure karein

Kisi bhi symbol se pehle, ek flat sheet of paper dekho. Uspar ek cheez rakho — maano ek billi. Ab pucho: "billi kahan hai?"

Aap jawab nahi de sakte bina ek doosre mark ke: ek fixed reference dot jis par sab log agree karein. Hum us dot ko origin kehte hain aur use $O$ se label karte hain.

YEH KYUN ZAROORI HAI: number "3" ka koi matlab nahi. "$O$ ke dayi taraf 3 metres" ek real jagah hai. Location sirf kisi reference ke relative mein exist karti hai. Dekho Coordinate systems and unit vectors.

1. Axes, coordinates, aur char quadrants

$O$ se hum right angles par do rulers bichchate hain: ek dayi taraf jaata hai (x-axis), ek upar jaata hai (y-axis). Teesra, page se aapki taraf aata hai, z-axis hai — flat problems mein hum ise mostly ignore karte hain.

Zaroori baat yeh hai ki har ruler dono directions mein jaata hai. Dayi chalna positive $x$ hai; bayi chalna negative $x$ hai. Upar chalna positive $y$ hai; neeche chalna negative $y$ hai. Toh ek coordinate ek signed number hai: sign batata hai ki ruler ke kaunse direction mein.

Do axes flat plane ko chaar quadrants mein kaatते hain, I–IV ke naam se $(x,y)$ ke signs ke hisaab se: I hai $(+,+)$ top-right, II hai $(-,+)$ top-left, III hai $(-,-)$ bottom-left, IV hai $(+,-)$ bottom-right.

SIGNS KYUN MATTER KARTE HAIN: motion har direction mein jaati hai, sirf right-aur-up mein nahi. Negatives ke bina aap $O$ ke bayi ya neeche kisi point ko describe hi nahi kar sakte.

YEH FIX KYUN KAREIN: bina stated orientation ke, "page se bahar" ambiguous hai — koi z-axis ko flip kar sakta hai. Right-hand rule ise pin down karta hai.

COORDINATES KYUN: numbers se calculate karna pictures se aasaan hai. Coordinates geometry se arithmetic ka bridge hain.

2. "Vector" hota kya hai — length aur direction wala ek arrow

Yeh raha pure chapter ka star.

Contrast karo ek scalar se — bas ek plain number bina direction ke (jaise temperature, ya ek length). Yeh distinction Scalars and vectors ki spine hai.

Chota hat symbol $\hat{}$ abhi aane wala hai — ruko.

3. Unit vectors $\hat{i},\hat{j},\hat{k}$ — "one step" arrows

Kisi bhi arrow ko numbers se build karne ke liye, hume standard "one-step" arrows chahiye jisse build karein.

YEH KYUN: plane par koi bhi arrow "itne $\hat{i}$'s dayi, itne $\hat{j}$'s upar" se reach kiya ja sakta hai. Steps ki negative ginti bas ulti direction ka matlab hai — jaise $-2\hat{i}$ do steps bayi hai. Toh ek generic arrow $\vec{v}$ ko $\vec{v}=x\hat{i}+y\hat{j}+z\hat{k}$ ke roop mein likhne ka literally matlab hai: "$O$ se shuru karo, $x$ steps x ke along lo, phir $y$ steps y ke along, phir $z$ steps z ke along," jahan negative components direction reverse karte hain. §1 se signed coordinates arrow ke components ban gaye hain. Aur details Coordinate systems and unit vectors mein.

4. Position vector $\vec{r}$ — "yahan" tak ka arrow

Ab hum topic ki pehli badi object naam de sakte hain. Yahan se aage, specific symbol $\vec{r}$ iske liye reserved hai.

Symbol $\vec{r}$ bas ek naam hai — "$r$" for radius/reach from the origin, upar vector-arrow ke saath taaki aap kabhi na bhoolo ki iska direction hai.

YEH KYUN IMPORTANT HAI: zero vector woh ek case hai jahan "arrow kaunsi taraf point karta hai?" ka koi jawab nahi — aur yeh exactly kisi bhi round trip ka displacement hai jo apne start par wapas aata hai.

5. Magnitude $|\vec{r}|$ aur Pythagoras engine

Aksar hum sirf arrow ki length chahte hain — point $O$ se kitni door hai — direction chhod ke.

Hum components se length kaise paate hain? Components $x$ (across) aur $y$ (up) ek right triangle ki do legs hain, aur arrow $\vec{r}$ uski slanted side (hypotenuse) hai. Kyunki legs square hoti hain, koi bhi negative signs gayab ho jaate hain — length positive aati hai chahe P kisi bhi quadrant mein ho.

YEH TOOL KYUN AUR DOOSRA NAHI? Hamare paas do perpendicular legs hain aur slanted length chahiye. Wahi ek rule jo right triangle ki do legs ko uske slant se connect karta hai woh hai Pythagoras theorem — "slant ka square legs ke squares ke sum ke barabar hai." Yeh exactly woh sawaal hai jo hum pooch rahe hain, toh yahi tool hum uthate hain. Square root bas "squared" ko undo karta hai taaki plain length wapas mile.

6. Symbol $\Delta$ — "change in"

Chote letters $i$ aur $f$ jodo (${\vec{r}}_i$, ${\vec{r}}_f$) ka matlab hai initial (start) aur final (end). Toh $\Delta \vec{r}={\vec{r}}_f-{\vec{r}}_i$ padha jaata hai "position mein change = end arrow minus start arrow."

YEH KYUN CHAHIYE: motion change ke baare mein hai, absolute location ke baare mein nahi. $\Delta$ compact tarika hai yeh kehne ka ki "yeh kitna shift hua?"

7. Arrows subtract karna — ${\vec{r}}_f-{\vec{r}}_i$ ke peeche ki picture

Displacement ki puri idea ek operation par tikhi hai: do arrows subtract karna. Yeh dikhta kaise hai yahan batate hain.

${\vec{r}}_f-{\vec{r}}_i=(x_f-x_i)\hat{i}+(y_f-y_i)\hat{j}$

Yeh Vector addition and subtraction ki machinery hai. Notice karo ki origin $O$ dono ${\vec{r}}_i$ aur ${\vec{r}}_f$ ke andar dikhta hai — toh jab aap subtract karte ho, woh cancel ho jaata hai. Yahi wajah hai ki displacement ko parwah nahi ki aapne $O$ kahan rakha. Agar start aur end coincide karein, toh yeh subtraction $\vec{0}$ deta hai — §4 se zero vector.

8. Summation $\sum$ — raasta jodna

Distance ke liye hum path ke har piece ki lengths jodте hain.

KYUN: path kitni bhi baar bend kar sakta hai; $\sum$ tidy tarika hai yeh kehne ka "jodते raho, chahe kitne bhi pieces hon." Har piece ki length Pythagoras (§5) se phir milti hai.

9. Inequality symbol $\le$

YEH KYUN CHAHIYE (aur sirf $<$ ya $=$ kyun nahi): topic ka headline result straight gap ko chale gaye raaste se compare karta hai, aur yeh do kabhi kabhi equal hote hain (ek seedha, one-way walk) lekin usually raasta zyada lamba hota hai (koi bhi bend ya reversal). Ek single symbol dono cases cover kare — exactly yahi $\le$ karta hai. Plain $<$ use karna galat hoga kyunki straight-line tie forbid ho jaata; $=$ use karna galat hoga kyunki har bent path forbid ho jaata.

Topic ka headline result $|\Delta \vec{r}|\le \text{distance}$ padhta hai: "straight-line gap kabhi bhi chale gaye raaste se lamba nahi hota — best case mein dono tie karte hain." Woh sirf tab tie karte hain jab aap kabhi na mudein ya double back karein — woh special case Motion in a straight line mein padha jaata hai.

10. Vocabulary ko kaam mein lagana (mini example)

Yeh seedha Average velocity and average speed se jodta hai: velocity $\Delta \vec{r}$ se banti hai, speed distance se.

Equipment checklist

Khud test karo — reveal karne se pehle jawab zor se bolo.