Intuition The one core ideaA 3D rotation is fully described by two things: a direction to spin around (an arrow) and how much to twist (an angle). A quaternion is just a tidy box of 4 numbers that stores both at once, kept honest by one rule — the four numbers squared must add to exactly 1.
Everything below is the vocabulary you need before that sentence makes full sense: what an axis is, what an angle is, what "squared and added to 1" means geometrically, and why 4 numbers describe a 3-way freedom.
This page assumes nothing . We build every symbol the parent note (parent topic) uses, one at a time, each anchored to a picture.
Start with the simplest object: a single number, like 3 − 2 point on a line . That is a scalar — a lone number with no direction, just a size (and a sign).
Now stack three numbers together: ( x , y , z ) arrow starting at the origin and reaching the point x y z vector . We draw a little hat or arrow on top to remember it points somewhere: v
Definition Scalar vs vector
Scalar s magnitude (size) only.Vector v = ( x , y , z ) magnitude and direction .
Why the topic needs both: a rotation carries a scalar (how much you twist) AND a vector (which way the spin-axis points). The quaternion q = ( q 0 , q v )
Look at the arrow again. How long is it? If it goes x y x y
Intuition Why we square, add, and square-rootA right triangle obeys ( leg ) 2 + ( leg ) 2 = ( hypotenuse ) 2 x 2 + y 2 z Squaring removes negative signs (a length can't be negative and left-vs-right shouldn't matter); the square root undoes the squaring to return an honest length.
A unit vector is an arrow of length exactly 1 , written with a hat: n ^ direction only — it points a way without adding size.
To make any arrow a unit vector, divide it by its own length ("normalize " it): v ^ = v / ∥ v ∥
Why the topic needs it: a spin-axis is a pure direction — "spin about this way." So the axis n ^ = ( n x , n y , n z ) n x 2 + n y 2 + n z 2 = 1
An angle θ how far you turn . Picture a clock hand sweeping from one position to another. A full turn is 36 0 ∘ 2 π
Definition Sine and cosine, on the circleDraw a unit circle (radius 1). Sweep a point around it by angle θ
cos θ horizontal coordinate.sin θ vertical coordinate.
Because the point sits on a radius-1 circle, Pythagoras gives the identity you will use constantly:
cos 2 θ + sin 2 θ = 1.
The quaternion product formula uses two vector products. Meet them both.
Definition Dot product (scalar-valued)p ⋅ q = p x q x + p y q y + p z q z number (a scalar).
Picture: how much two arrows point the same way . Same direction → big positive; perpendicular → zero; opposite → negative.
Definition Cross product (vector-valued)p × q = a new arrow perpendicular to both p and q . vector , sticking out sideways from the plane of the two inputs.
Picture: point your right-hand fingers from p q p × q
You may have met one imaginary unit before: i i 2 = − 1 Complex Numbers as 2D Rotations ). Quaternions have three of them: i , j , k
Definition The imaginary unitsEach squares to − 1 i 2 = j 2 = k 2 = − 1. ij = k , j k = i , k i = j Picture: think of i , j , k
The full quaternion is then written q = q 0 + q 1 i + q 2 j + q 3 k q = ( q 0 , q v ) q v = ( q 1 , q 2 , q 3 ) q 0 i , j , k
Common mistake "The vector part( q 1 , q 2 , q 3 ) Why it feels right: both are 3-number bundles.
Fix: the axis n ^ unit vector; the quaternion's vector part is n ^ sin 2 θ scaled by sin 2 θ
Before we can even say "apply a rotation," we need the operations that act on quaternions. Meet them now — nothing later uses them until they are defined here.
Definition The Hamilton product⊗ ⊗ multiply two quaternions . It is NOT ordinary number multiplication — it mixes the scalar and vector parts using the dot and cross products from Section 4:
p ⊗ q = ( p 0 q 0 − p ⋅ q , p 0 q + q 0 p + p × q ) . Picture: feed in two boxes-of-4, get one box-of-4 out. The p ⋅ q p × q flips sign when you swap the inputs (p × q = − q × p non-commutative : in general p ⊗ q = q ⊗ p
q ∗ conjugate flips the sign of the vector part only:
q ∗ = ( q 0 , − q v ) = ( q 0 , − q 1 , − q 2 , − q 3 ) . Picture: keep the twist-amount, but point the spin-axis the opposite way — this is the undo button for a rotation.
q − 1 inverse is the quaternion that, multiplied in, gives back the "do-nothing" identity ( 1 , 0 ) q − 1 = ∥ q ∥ 2 q ∗ . For a unit quaternion (∥ q ∥ = 1 1 q − 1 = q ∗ (just negate the 3 vector signs). Why this matters: inverting a rotation should be cheap. With unit quaternions it is just three sign flips — no division, no matrix work. This is why spacecraft flight computers (Spacecraft Attitude Determination ) love them.
Now that ⊗ q − 1 see where the mysterious θ /2
Intuition The rotation sandwichTo actually rotate an arrow v sandwich it:
v ′ = q ⊗ v ⊗ q − 1 . v on both sides , so its rotation effect is applied twice . If a single q θ 2 θ q half the angle, θ /2 θ
Now every symbol in the headline formula is defined. Read it slowly:
Intuition Why 4 numbers, 1 rule = 3 freedoms — the 3-sphere pictureFour numbers ( q 0 , q 1 , q 2 , q 3 ) ∑ q i 2 = 1 4 − 1 = 3 point pinned to the surface of a sphere , always at distance 1 from the centre. A surface has one fewer dimension than the space it sits in — a 2D sphere-surface lives in 3D, and our "3-sphere" surface lives in 4D. You can't draw 4D, so the figure below shows the honest 3D stand-in: an ordinary ball whose surface is where every valid rotation lives.
Definition The two signs are the same rotation (double cover)Look at the sandwich v ′ = q ⊗ v ⊗ q − 1 q − q both sides and cancel :
( − q ) ⊗ v ⊗ ( − q ) − 1 = q ⊗ v ⊗ q − 1 . q − q identical 3D rotation. Every real rotation is therefore named by two quaternion points (antipodal on the 3-sphere). This "two-names-one-rotation" fact is called the double cover of S O ( 3 )
q − q Why it feels right: they are different 4-tuples, so surely different outputs.
Fix: the sandwich squares the sign away. − q other name for the same physical turn.
θ = 0 zero rotation , q = ( cos 0 , n ^ sin 0 ) = ( 1 , 0 ) n ^ has no effect and is undefined — which is correct: "spin by nothing" doesn't care about which axis you didn't spin around. This ( 1 , 0 ) identity quaternion , the do-nothing element.
θ = 18 0 ∘ π cos 2 π = 0 q = ( 0 , n ^ ) pure vector quaternion, scalar part zero. Here q − q n ^ − n ^ 18 0 ∘ n ^ 18 0 ∘ − n ^
Quaternion q = q0 plus vector part
cos and sin on unit circle
Cross product - a new arrow
Hamilton product times operator
Double cover q and minus q
Unit constraint sum of squares = 1
Represents a real 3D rotation
This map feeds directly into Axis-Angle Representation , Rotation Matrices SO(3) , Euler Angles and Gimbal Lock , and later Quaternion Kinematics — dq/dt and Spacecraft Attitude Determination .
Cover the right side and test yourself — you're ready for the parent note when all reveal cleanly.
A scalar is... a single number; picture = a point on a line, magnitude only.
A vector ( x , y , z ) an arrow from the origin; magnitude and direction.
Length of v = ( x , y , z ) ∥ v ∥ = x 2 + y 2 + z 2 (3D Pythagoras).
A unit vector is... an arrow of length exactly 1, written n ^
To normalize a vector you... divide it by its own length,
v ^ = v / ∥ v ∥ .
cos θ sin θ the horizontal and vertical coordinates of a point swept angle θ
The identity linking them is... cos 2 θ + sin 2 θ = 1
The dot product p ⋅ q a number : p x q x + p y q y + p z q z
The cross product p × q a vector perpendicular to both; flips sign when you swap order.
The units obey... i 2 = j 2 = k 2 = − 1 ij = k , j k = i , k i = j
The ⊗ the Hamilton (quaternion) product; non-commutative; scalar part
p 0 q 0 − p ⋅ q , vector part
p 0 q + q 0 p + p × q .
The conjugate q ∗ ( q 0 , − q v ) — flip the vector part; the undo direction.
The inverse of a unit quaternion is... q − 1 = q ∗ ∥ q ∥ = 1
The rotation sandwich is... v ′ = q ⊗ v ⊗ q − 1 q
Why the half-angle θ /2 the two-sided sandwich applies q
q − q the same rotation (double cover); signs cancel in the sandwich.
At θ = 0 ( 1 , 0 ) , the identity; the axis is undefined (spinning by nothing).
At θ = 18 0 ∘ q 0 = cos 9 0 ∘ = 0
The vector part vs the axis... vector part = n ^ sin 2 θ scaled ), not the bare axis.
Why 4 numbers, 1 rule... 4 − 1 = 3
