Plane change maneuvers — Δv = 2v·sin(Δi - 2): Foundations

Before you can read $Δ v = 2 v sin (Δ i /2)$ you need to know, honestly and from zero, what each mark means: what a vector is, what its magnitude $v$ is, what an angle $Δ i$ is, what $sin$ asks, what the funny $Δ$ means, and how the law of cosines stitches them together. We build them in that order — each one leaning on the one before.

1. The arrow: what a vector is

A plain number like "7" is called a scalar — it has size only. Temperature is a scalar. But velocity is a vector: "moving at 7 km/s" is incomplete — 7 km/s which way?

Figure — Plane change maneuvers — Δv = 2v·sin(Δi - 2)

Look at the red arrow in the figure. Its tail is where it starts, its tip is where it points. Two facts live in one arrow: how long and which way. The parent topic is entirely about the second fact changing while the first stays fixed.

We write a vector with a little arrow on top: $v$ . The same letter without the arrow, $v$ , means its length only — coming up next.

2. The length of the arrow: magnitude $v$

So $v$ is the arrow; $v = ∣ v ∣$ is how long that arrow is. For a satellite, $v$ is its speed in km/s.

You'll meet the vis-viva equation in Orbital velocity — vis-viva equation — that's the machine that tells you the actual number $v$ to plug in here.

3. Turning arrows and the meaning of an angle $Δ i$

In the figure the red arc is the angle between the two black arrows. A tiny arc = a small turn; a big arc = a big turn. That arc is the entire cost driver of a plane change.

The symbol $Δ$ — "change in"

So there are two deltas on the parent page and they are different animals:

$Δ i$ — a change in an angle (how much you tilted the plane), measured in degrees.
$Δ v$ — a change in a vector (the burn you must fire), measured in km/s.

The letter $i$ itself is the inclination — the tilt angle of the orbital plane against the equator. That's the job of Inclination and orbital elements; here we only need " $i$ is an angle, $Δ i$ is how much it changed."

4. The sine question: what $sin$ asks

Angles are hard to compute with directly, so we translate them into ratios of lengths using a right triangle.

Take a right triangle with one chosen angle $θ$ (Greek "theta", a stand-in name for any angle). Name its sides relative to $θ$ :

opposite — the side across from $θ$ (red in the figure),
hypotenuse — the long slanted side facing the square corner.

A few values worth carrying in your pocket (each is a length-ratio you can read off a standard triangle):

$θ$	$sin θ$	meaning
$0°$	$0$	no turn, no height
$30°$	$0.5$	rises half the hypotenuse
$90°$	$1$	rises the full hypotenuse

Recall Quick check on sine

If $θ = 30°$ , what fraction of the hypotenuse is the opposite side? $sin 30° = 0.5 :::$ exactly half — this is the value that makes a $60°$ plane change cost $Δ v = v$ .

5. Cosine and the law of cosines: the engine

Cosine is sine's partner: instead of the opposite side, it uses the side next to the angle.

The parent derivation runs on the law of cosines, which is Pythagoras upgraded to work for any triangle, not just right-angled ones.

Setting $a = b = v$ and $C = Δ i$ hands you the parent's Step 1 directly: $Δ v^{2} = v^{2} + v^{2} - 2 v v cos Δ i = 2 v^{2} (1 - cos Δ i)$

6. Putting arrows together: vector subtraction

$Δ v = v_{2} - v_{1}$ is a subtraction of arrows. Here is the picture, once and for all.

To subtract, place the two velocity arrows tail-to-tail. The red arrow that runs from the tip of $v_{1}$ to the tip of $v_{2}$ is $Δ v$ — the burn you must fire. Notice: even though both black arrows are the same length $v$ , the red connecting arrow is clearly not zero. That single picture kills the " $Δ v = 0$ " trap.

v

mean vs plain

v

v

is the velocity vector (length + direction);

v = ∣ v ∣

is only its length (the speed)

What does

Δ

read as?

"the change in" — final minus initial

Why are

Δ i

and

Δ v

different kinds of thing?

Δ i

is a change in an angle (degrees);

Δ v

is a change in a vector (a burn, km/s)

Define

sin θ

on a right triangle.

opposite side divided by hypotenuse

Why is sine the right tool for this topic?

it converts a turn (angle) into a length (the half-base of the velocity triangle), which is exactly what

Δ v

sin 30° = ?

and why does it matter?

0.5

— it makes a

60°

plane change cost

Δ v = v

State the law of cosines.

c^{2} = a^{2} + b^{2} - 2 ab cos C

for sides

a, b

and included angle

C

What does law of cosines reduce to when

C = 90°

plain Pythagoras

c^{2} = a^{2} + b^{2}

How do you subtract two arrows

v_{2} - v_{1}

tail-to-tail, draw the arrow from tip of

v_{1}

to tip of

v_{2}

Why isn't

Δ v

zero when both speeds equal

v

subtraction is of vectors; the connecting arrow has real length even with equal-length sides

Why does the half angle appear?

perpendicular from the apex of the isosceles triangle bisects both the angle

Δ i

and the base

Why do engineers fear large

Δ v

Tsiolkovsky makes fuel grow exponentially,

m_{0} / m_{f} = e^{Δ v / v_{e}}

Connections

Parent topic (Hinglish)
Orbital velocity — vis-viva equation (supplies the number $v$ )
Inclination and orbital elements (defines $i$ and $Δ i$ )
Vector addition and law of cosines (the math engine here)
Tsiolkovsky rocket equation (turns $Δ v$ into fuel)
Apoapsis and periapsis (where $v$ is smallest — cheapest to turn)
Hohmann transfer orbit (where plane changes get combined)

Foundations — Plane change maneuvers — Δv = 2v·sin(Δi - 2)