Some quantities need only a number: a temperature of 20°, a mass of 3 kg. Others need a number and a direction: "go 5 metres — but which way?"
Figure s01 below shows a single vector drawn twice in two places.
Look at Figure s01. The cyan arrow is A. Slide it around without turning it (the amber copy) and it is still the same vector — only length and direction matter, not where you draw it. This tiny fact is the whole reason a spinning wheel can act as a compass: its angular-momentum arrow stays put in space.
Before we multiply anything by a vector we must say what that even means, and what it means to add two arrows together. (We will meet the tiny arrow dL later, in §5, after we have defined what the little d means — so for now we add two ordinary arrows.)
Figure s04 shows the head-to-tail rule for the special case we will need most: a big arrow with a tiny arrow stuck on its tip at a right angle.
Look at Figure s04. Because the little added arrow (amber) is stuck onto the tip of the big cyan arrow at a right angle, the new dashed arrow has almost the same length but a slightly turned direction. Hold that picture — it is the entire secret of precession, which we will unlock once the d notation arrives in §5.
A car's speedometer measures how fast it moves forward. For spinning we need a different meter: how fast the object turns.
But "spinning" secretly carries a direction too: which axis, and which way around it. So we promote the scalar to a vector.
Why radians and not degrees? Radians are the "natural" angle unit where arc-length = radius × angle with no extra conversion factor. That clean relationship is exactly what we'll use later to compute how far the tip of an arrow slides sideways. Degrees would smuggle in an ugly π/180.
To get a heavy shopping trolley moving you push hard; its mass resists changes in straight-line motion. Spinning has its own version of "heavy to get going."
Figure s02 compares two wheels of equal mass but different mass placement.
In Figure s02 the two wheels have the same mass, but the right one has its mass pushed out to the rim. That one has a larger I — it is the better gyroscope, because faraway mass resists spin changes more strongly.
The parent note's whole derivation rests on one expression: τ=dtdL. Let us defuse that fraction.
The updated spin-arrow is the head-to-tail sumL+dL — this is exactly the picture in Figure s04 from §0b, now with real names on the arrows. For a tipping torque, dL points the way the torque points, which is sideways to L — so the sum has almost the same length but a turned direction. Lturns instead of falling. That is the seed of precession.
The parent quotes τ=Ω×L. We will not just quote it — we will build it. First the tool.
Figure s06 shows the angle θ measured tail-to-tail and the perpendicular output arrow.
Now meet the second turning-rate. The wheel spins fast about its own axis at rate ω; but the whole axis also slowly walks around in a circle. That slow walk is its own rotation, with its own angular-velocity vector.
How a rotating vector changes — the key link. Suppose L keeps its length fixed but its whole arrow is being rotated by the precession Ω. What is dL/dt?
So the tip of L, dragged around by Ω, moves at velocity
dtdL=Ω×L.
Read it out: the change in L is perpendicular to both Ω and L (a sideways slide), with size ΩLsinθ — exactly a rotating arrow. Now combine with the one law τ=dL/dt from §5:
Figure s05 shows the tip of L tracing a circle, with dL/dt=Ω×L tangent to it.
The diagram below is a dependency chart: an arrow "X→Y" means "you need X before Y makes sense." Start at the top boxes (pure ideas that need nothing) and follow the arrows down; every path eventually funnels into the gyroscope topic at the bottom. Reading it top-to-bottom is the exact order this page introduced the symbols, and shows why each one had to come before the next.
