Intuition The ONE core ideaA rocket's motion is described by arrows in space (positions, velocities), but the numbers we write for those arrows depend on which set of axes we stand on — and different jobs need different axes. Everything in this topic is just the same arrow, re-measured against a rotated set of rulers , so the whole subject reduces to turning one set of axes into another .
Before you can read a single formula in the parent note, you need to own every symbol it throws at you. We build them one at a time, each earning its place before the next arrives.
Definition Vector (an arrow) — writtenr r directed quantity ": it has a length and a direction . Picture a physical stick pointing from Earth's centre to the rocket. r
Definition Reference frame (a set of rulers)A frame is three rulers glued together at right angles , meeting at a chosen point called the origin . To "measure" an arrow you read off how far along each ruler it reaches. Those three readings are the arrow's components in that frame.
Intuition The whole subject in one sentenceThe arrow r
If I lay down rulers labelled x , y , z r x x r y , r z column of three numbers :
r = r x r y r z same arrow measured on different rulers gives a different list — that difference is the entire game.
Worked example Same arrow, two framesAn arrow pointing straight "East" reads ( 0 , 5 , 0 ) y x ( 5 , 0 , 0 )
Definition Unit vector (the "hat" symbol)A hat ^ 1 x ^ x Z ^ r ^ radial unit vector) is a 1-metre stick pointing straight out from Earth's centre through your feet.
Picture: stand on the ground. r ^ − r ^
Intuition Why hats matter hereThe parent note builds the North / East / Down rulers out of r ^ Z ^ directions with no length attached — pure "which way", which is exactly what a ruler axis is.
Definition Right-handed setPoint the fingers of your right hand along x ^ y ^ thumb then points along z ^ sign of the third axis so nobody accidentally builds a mirror-image world.
Common mistake Ignoring handednessWhy it feels harmless: "x y z The trap: a left-handed set flips one sign in every cross product, so your "East" comes out pointing West. Fix: always check with the right hand.
Angles say how much one thing is turned from another . Each Greek letter has a fixed job:
Definition The five angles used in the parent note
θ ϕ latitude : how far North (up from the equator) you stand, 0° 90° λ longitude : how far East around the equator, 0° ψ yaw , θ pitch , φ roll : the three ways a rocket can twist.
Mnemonic Latitude vs longitudeLat itude are the flat ladder-rungs (climb North). Long itude are the long orange-slice lines running pole to pole.
Intuition Why trig shows up everywhere hereEvery rotation "splits" an arrow into a part along one axis and a part along a perpendicular axis. Cosine measures the along-part (how much survives in the same direction); sine measures the sideways-part (how much leaks into the perpendicular direction). We need trig because that split is exactly what changing rulers does.
cos sin 1 x θ shadow on the horizontal axis is cos θ shadow on the vertical axis is sin θ
At θ = 0 x cos 0 = 1 sin 0 = 0
At θ = 90° cos 90° = 0 sin 90° = 1
This is why Example 2 in the parent note says "with θ ≈ 90° x E ≈ 0
Intuition Why we need this toolTo read a component, I ask: how much of my arrow points along this ruler? The dot product answers exactly that single question — no other tool gives you "the overlap" between two directions in one number.
a ⋅ b a ⋅ b = a x b x + a y b y + a z b z ∣ a ∣ ∣ b ∣ cos α α Picture: if the two arrows point the same way, cos 0 = 1 cos 90° = 0 unit ruler direction.
Intuition Why a second product?Sometimes I don't want overlap — I want a brand-new direction at right angles to two others . That is the cross product's unique job, and it is how the parent note builds "East = Z ^ × r ^
a × b perpendicular to both a b a → b ∣ a ∣∣ b ∣ sin α Picture: Z ^ r ^ East — the direction Earth carries you.
Definition Matrix (a table that transforms a list)A 3 × 3 R new column of three — the same arrow read on rotated rulers.
R T R = I R new ruler directions written in the old frame. R T R = I
ω e ω e how fast Earth spins , measured in radians per second: ω e ≈ 7.292 × 1 0 − 5 rad/s ≈ 15° t angle turned so far : θ = ω e t only thing separating ECI from ECEF.
A radian is the "natural" unit of angle: a full circle is 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
See the parent map here: parent topic .
Dot & cross products, right-handedness → building the NED rulers and the transport theorem: Rotating Reference Frames and Coriolis Force .Rotation matrix R R T det R → the rotation chain: Rotation Matrices and Euler Angles , and the alternative in Quaternions for Attitude .Inertial vs non-inertial frames → why we integrate in ECI: Newton's Laws and Inertial Frames .Latitude ϕ → the cos ϕ Launch Azimuth and Orbital Inclination , and the fine print in Geodetic vs Geocentric Latitude .The full equations these frames carry → Rocket Equations of Motion .
Recall Self-test: can you answer each before reading the parent note?What does the hat in 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
