3.4.1 · D1Coordinate systems — Earth-Centered Inertial (ECI), Earth-Ce
Ek rocket ki motion ko space mein arrows (positions, velocities) se describe karte hain, lekin un arrows ke liye jo numbers hum likhte hain woh depend karte hain kis set of axes pe hum khade hain — aur alag-alag kaam ke liye alag-alag axes chahiye. Is topic mein sab kuch bas wahi ek arrow hai, ek rotated set of rulers ke against re-measure kiya gaya , isliye poora subject reduce ho jaata hai ek set of axes ko doosre mein badalne mein.
Parent note mein ek bhi formula padhne se pehle, tumhe uske har symbol ka ownership lena hoga. Hum unhe ek-ek karke build karte hain, har ek apni jagah earn karta hai agle ke aane se pehle.
Definition Vector (ek arrow) — likha jaata hair r directed quantity hai": iska ek length hai aur ek direction hai. Ek physical stick imagine karo jo Earth ke centre se rocket ki taraf point karti hai. r
Definition Reference frame (rulers ka ek set)Ek frame teen rulers hain jo right angles pe ek saath jude hain , ek chosen point pe milte hain jise origin kehte hain. Kisi arrow ko "measure" karne ke liye tum padhte ho har ruler ke along woh kitni door tak pahuncha . Woh teen readings arrow ke components hain us frame mein.
Intuition Poora subject ek sentence meinArrow r
Agar main x , y , z r x x r y , r z teen numbers ke column mein stack karte hain:
r = r x r y r z Wahi arrow alag rulers pe measure kiya toh alag list milti hai — wahi difference poora game hai.
Worked example Wahi arrow, do framesEk arrow jo straight "East" point kar raha hai woh ( 0 , 5 , 0 ) y x ( 5 , 0 , 0 )
Definition Unit vector ("hat" symbol)Hat ^ 1 x ^ x Z ^ r ^ radial unit vector kehte hain) ek 1-metre stick hai jo straight out point karta hai Earth ke centre se tumhare paon ke through.
Picture: zameen pe khade ho. r ^ − r ^
Intuition Hats yahan kyun matter karte hainParent note North / East / Down rulers ko r ^ Z ^ directions hain jinse koi length attached nahi — pure "which way", jo exactly wahi hai jo ek ruler axis hota hai.
Definition Right-handed setApne right haath ki ungliyan x ^ y ^ thumb phir z ^ sign fix karta hai taaki koi accidentally mirror-image world build na kar le.
Common mistake Handedness ko ignore karnaKyun yeh harmless lagta hai: "x y z Trap: ek left-handed set har cross product mein ek sign flip kar deta hai, toh tumhara "East" West ki taraf point karne lagta hai. Fix: hamesha right hand se check karo.
Angles batate hain ek cheez doosri se kitni turn hai . Har Greek letter ka ek fixed kaam hai:
Definition Parent note mein use hone wale paanch angles
θ ϕ latitude : tum kitne North (equator se upar) khade ho, 0° 90° λ longitude : equator ke around kitna East, 0° ψ yaw , θ pitch , φ roll : teen tarike jinse ek rocket twist kar sakta hai.
Mnemonic Latitude vs longitudeLat itude woh flat ladder-rungs hain (North climb karo). Long itude woh long orange-slice lines hain jo pole se pole tak chalti hain.
Intuition Trig yahan har jagah kyun aata haiHar rotation ek arrow ko ek axis ke along part mein aur ek perpendicular axis ke along part mein "split" karta hai. Cosine along-part measure karta hai (same direction mein kitna bachta hai); sine sideways-part measure karta hai (perpendicular direction mein kitna jaata hai). Hum trig isliye use karte hain kyunki wahi split exactly wahi hai jo rulers badalna karta hai.
Definition Unit circle pecos sin 1 x θ Horizontal axis pe uska shadow hai cos θ vertical axis pe uska shadow hai sin θ
θ = 0 x cos 0 = 1 sin 0 = 0 θ = 90° cos 90° = 0 sin 90° = 1
Isliye parent note ka Example 2 kehta hai "θ ≈ 90° x E ≈ 0
Intuition Hum yeh tool kyun chahiyeComponent read karne ke liye main poochhta hun: mera arrow is ruler ki taraf kitna point karta hai? Dot product exactly usi ek sawaal ka jawaab deta hai — koi aur tool tumhe do directions ke beech "overlap" ek number mein nahi deta.
a ⋅ b a ⋅ b = a x b x + a y b y + a z b z ∣ a ∣ ∣ b ∣ cos α α Picture: agar do arrows same way point karein, cos 0 = 1 cos 90° = 0 unit ruler direction ke saath dot karna.
Intuition Doosra product kyun?Kabhi kabhi mujhe overlap nahi chahiye — mujhe ek brand-new direction chahiye jo do doosron ke right angles pe ho . Yahi cross product ka unique kaam hai, aur isi se parent note "East = Z ^ × r ^
a × b dono a b ho, uski direction right hand se set hoti hai (fingers a → b ∣ a ∣∣ b ∣ sin α Picture: Z ^ r ^ East ki taraf point kare — woh direction jisme Earth tumhe carry karta hai.
Definition Matrix (ek table jo ek list transform karti hai)Ek 3 × 3 R naya column return karta hai — wahi arrow rotated rulers pe padha gaya.
R T R = I R new ruler directions hain jo old frame mein likhi gayi hain. R T R = I
ω e ω e Earth kitni tez spin karta hai , radians per second mein measure kiya: ω e ≈ 7.292 × 1 0 − 5 rad/s ≈ 15° t ab tak kitna angle turn hua : θ = ω e t ek hi cheez hai.
Radian angle ki "natural" unit hai: ek poora circle 2 π ≈ 6.283 sin cos
Unit vectors x hat Z hat r hat
Frame transforms ECI ECEF NED body
Coordinate systems topic 3.4.1
Parent map yahan dekho: parent topic .
Dot & cross products, right-handedness → NED rulers aur transport theorem build karna: Rotating Reference Frames and Coriolis Force .Rotation matrix R R T det R → rotation chain: Rotation Matrices and Euler Angles , aur alternative Quaternions for Attitude mein.Inertial vs non-inertial frames → kyun hum ECI mein integrate karte hain: Newton's Laws and Inertial Frames .Latitude ϕ → cos ϕ Launch Azimuth and Orbital Inclination , aur fine print Geodetic vs Geocentric Latitude mein.Woh poori equations jo yeh frames carry karti hain → Rocket Equations of Motion .
Recall Self-test: kya tum parent note padhne se pehle har ek ka jawaab de sakte ho?r ^ R R − 1 = R T R T R = I a ⋅ b = ∣ a ∣∣ b ∣ cos α a × b cos θ θ θ = ω e t t ϕ ϕ λ
