Rotation Matrices

Equivalent forms

R^{\top}R = I,\quad \det R = 1

R(\alpha)R(\beta) = R(\alpha+\beta)

Composition of rotations is addition of angles — the matrix product R(α)R(β)=R(α+β) turns geometry into arithmetic.

Symbol	Name	SI unit	Dimension	Range
$\theta$	Rotation angle Angle of rotation	rad	1	-3.1416 – 3.1416
$r$	Radius Distance of points from the origin	1	1	0.3 – 2
$n$	Point count Number of points in the cloud	1	1	3 – 12
$\det R$	Determinantoutput Always 1 for a proper rotation	1	1	1 – 1

Unit systems

Symbol	SI	CGS	Imperial
$\theta$	rad	rad	rad

Where it holds

SO(2) here; extends to SO(3) (three-parameter) and SO(n). Improper rotations

(det = -1) add

a reflection.

Dimensional analysis

R v (dim of v)

rotated v (dim of v)

R is dimensionless (pure cos/sin), so it preserves the dimensions and magnitude of any vector it acts on.

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Discovery

Euler · 1775

Euler's rotation theorem (1775) showed any 3D orientation is a single rotation about some axis. The matrix form and the group SO(3) matured in the 19th century; today rotations underpin robotics, graphics, spacecraft attitude, and the theory of angular momentum in quantum mechanics.

Try this

How does a game engine spin a spaceship, or a robot arm know where its hand is? One compact matrix rotates every point of space by the same angle at once — rigidly, reversibly, without ever stretching.

Turn the angle dial and watch a whole cloud of points swing around the origin together, each keeping its distance from the center.

Research status: stable

Real-world applications

3D graphics and game engines
Robot kinematics and CNC machining
Spacecraft and drone attitude control
Angular-momentum operators in quantum mechanics

Common misconceptions

3D rotations commute — they don't; order matters (unlike 2D).
A rotation can change a vector's length — it can't; that's the defining property.
$det = 1$ is automatic for orthogonal matrices — reflections are orthogonal with $det = -1$ .

Experimental verification

Robotic arms and CNC machines position tools to microns using chained rotation matrices; spacecraft star-trackers compute attitude as a rotation matrix and match predicted star positions.

Derivation

Require R preserve the dot product: Rᵀ

R = I

(orthogonal) and preserve orientation:

det R = +1

.

In 2D the general orthogonal, orientation-preserving matrix is exactly

[[\cos \theta

,-

\sin \theta ]

,

[\sin \theta

,

\cos \theta ]]

— its first column is a unit vector at angle

\theta

, the second its perpendicular.

Limiting cases

\theta = 0

⟶

R = I

No rotation; every point stays put.

\theta = \pi

⟶ point reflectionA half-turn sends every point to its opposite.

\theta = 2\pi

⟶ back to IFull turn is the identity — rotations are periodic.

What if…

What if

det R = -1

instead of +1?

It's an improper rotation — a rotation combined with a reflection, which reverses chirality.

What if I rotate in 3D about a tilted axis?

Use Rodrigues' formula or an axis-angle/quaternion; three angles (Euler angles) parameterize SO(3).

1

Rotate (1,0) by 90°

Given ·

v:: (1,0)
\theta:: $\pi /2$

Find · R v

Steps

$\cos 90^{\circ}=0$ , $\sin 90^{\circ}=1$
$R = [[0$ ,-1],[1,0]]
R(1, $0) = (0$ ,1)

Result · (0,1)

2

Compose two rotations

Given ·

\alpha:: $30^{\circ}$
\beta:: $45^{\circ}$

Find · combined angle

Steps

$R(\alpha )R(\beta )=R(\alpha +\beta )$
$30^{\circ}+45^{\circ}$
$= 75^{\circ}$

Result ·

75^{\circ}

Determinant→Eigenvalue Equation→Euler's Formula→group axioms