Geometric Transformations: Matrices in R² & R³

How to Read a Transformation Matrix

The geometric effect of a matrix

A

is revealed by two pieces of information: what it does to the standard basis vectors (the columns) and what its determinant is.

Column

1

is the image of

\mathbf{e}_1 = (1, 0)

in

\mathbb{R}^2

or

(1, 0, 0)

in

\mathbb{R}^3

. Column

2

is the image of

\mathbf{e}_2

. The matrix maps the standard grid to the parallelogram (or parallelepiped) spanned by these column vectors.

The absolute value

|\det(A)|

measures how the transformation scales areas (in

\mathbb{R}^2

) or volumes (in

\mathbb{R}^3

). The sign of

\det(A)

indicates orientation: positive means the transformation preserves handedness, negative means it reverses it. An orthogonal matrix (

\det = \pm 1

) preserves all lengths and angles — it is a rigid motion.

Geometric Transformation Notation

Notation

Geometric Transformation Notation

Named matrices with their defining parameter hung in the subscript — an angle, an axis, a factor — so the letter says what kind of motion it is and the subscript says which one. Two sign conventions ride along, and one composition order that reads backwards. All catalogued among the linear algebra symbols.

The matrix bracket anatomy and

AB

juxtaposition come from matrix operation notation;

\det

and

|\det|

as a scale factor from determinant and determinant geometry notation;

[T]

and its columns from matrix representation notation;

\theta

,

\cos

and

\sin

from trigonometric notation.

R_\theta

,

S_k

,

P_W

rotation by theta, scaling by k, projection onto W

A naming scheme rather than three separate symbols: the letter names the kind of motion and the subscript supplies the parameter that pins down which one. The subscript's type varies with the letter — an angle for

R

, a number for

S

, a whole subspace for the projection matrix — and reading it wrongly is the usual way these get confused.

CasesIn

\mathbb{R}^3

the rotation needs an axis as well, so the subscript grows:

R_{z,\theta}

or

R_{\mathbf{n}}(\theta)

name the axis first and the angle second — the convention varies, and Rotations in R³ below fixes one.

Do not confuse

P

at its other jobs. The same letter is a projection here, a change-of-basis matrix elsewhere, and a permutation matrix in factorisations — three unrelated tenants distinguished only by what sits in the subscript.

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

the minus sign sits above the diagonal — counterclockwise

The placement of the single minus is the entire convention: with

-\sin\theta

in the upper right, positive

\theta

turns counterclockwise, the standard orientation this page uses. Transposing the matrix — equivalently negating

\theta

— reverses the direction.

CasesComputer graphics often uses the opposite sign because screen coordinates run

y

downward, so the same matrix appears with the minus below the diagonal and still called "rotation by

\theta

"; the convention is a property of the coordinate system, not of the formula.

Do not confuseThe rotation's inverse as something to be computed.

R_\theta^{-1} = R_{-\theta} = R_\theta^{T}

— the transpose already is the inverse, since a rotation is orthogonal, and inverting one by elimination is wasted effort.

BA

means "do

A

first"

the rightmost factor acts first

Composing transformations writes the matrices in the reverse of the order they happen — Combining Transformations below relies on it — because each acts on the column to its right. The notation inherits its direction from function composition, not from reading order.

CasesThe consequence is that

AB \neq BA

carries geometric meaning here: rotating then reflecting is a different motion from reflecting then rotating, and the notation's order is what records which was meant.

Do not confuseLeft-to-right execution. Writing

R_\theta S_k

and reading it as "rotate, then scale" reverses the actual sequence — the scaling happens first, and the mistake is invisible until the two operations fail to commute.

All linear algebra symbols →↑ Linear Transformations

Scaling

Uniform scaling multiplies every coordinate by the same factor:

T(\mathbf{x}) = c\mathbf{x}

, with matrix

cI

. When

c > 1

the transformation enlarges, when

0 < c < 1

it shrinks, and when

c < 0

it reflects through the origin and scales.

Non-uniform scaling stretches each axis independently. In

\mathbb{R}^2

,

T(x, y) = (c_1 x, c_2 y)

has matrix

\text{diag}(c_1, c_2)

. The horizontal axis is scaled by

c_1

and the vertical by

c_2

. A unit square maps to a rectangle with side lengths

|c_1|

and

|c_2|

.

The determinant is

c^n

for uniform scaling and

c_1 c_2

(or

c_1 c_2 c_3

) for non-uniform. When any scaling factor is zero, the transformation collapses that axis entirely and the determinant is zero.

In

\mathbb{R}^3

,

\text{diag}(c_1, c_2, c_3)

scales each coordinate axis independently. The unit cube maps to a rectangular box with side lengths

|c_1|

,

|c_2|

,

|c_3|

and volume

|c_1 c_2 c_3|

.

Rotations in R²

Rotation by angle

\theta

counterclockwise about the origin has matrix

Rotation Matrix 2D

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Learn more about this formula: Rotation Matrix 2D →

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The first column

R_\theta \mathbf{e}_1 = (\cos\theta, \sin\theta)

is the image of

(1, 0)

— the point on the unit circle at angle

\theta

. The second column

R_\theta \mathbf{e}_2 = (-\sin\theta, \cos\theta)

is the image of

(0, 1)

— the point at angle

\theta + 90°

.

The determinant is

\cos^2\theta + \sin^2\theta = 1

for every

\theta

: rotations preserve areas and orientation. The matrix is orthogonal:

R_\theta^T R_\theta = I

, so lengths and angles are preserved. The inverse is

R_\theta^{-1} = R_{-\theta} = R_\theta^T

— rotating backward by the same angle.

Rotations compose by adding angles:

R_\alpha R_\beta = R_{\alpha + \beta}

. This follows from the trigonometric addition formulas and corresponds to the fact that rotating by

\beta

then by

\alpha

is the same as rotating by

\alpha + \beta

.

Common cases:

R_{90°} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

,

R_{180°} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

,

R_{45°} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}

.

Rotations in R³

In three dimensions, a rotation is specified by an axis and an angle. The three coordinate-axis rotation matrices are:

Rotation Matrices 3D

R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}, \;\; R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}, \;\; R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Learn more about this formula: Rotation Matrices 3D →

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Rotation by

\theta

about the

z

-axis leaves the

z

-coordinate unchanged and rotates the

xy

-plane:

R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}

Rotations about the

x

-axis and

y

-axis have the same

2 \times 2

rotation block embedded in different positions:

R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}, \quad R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}

Every

3 \times 3

rotation matrix is orthogonal with determinant

+1

. The axis of rotation is the eigenvector with eigenvalue

1

— the direction that remains fixed. Any rotation in

\mathbb{R}^3

can be decomposed into rotations about the coordinate axes (Euler angles), though the decomposition is not unique.

Axis of rotation	Matrix	Fixed direction	Plane that rotates
x-axis: Rₓ(θ)	[1 0 0; 0 cosθ −sinθ; 0 sinθ cosθ]	x-coordinate unchanged	yz-plane
y-axis: R_y(θ)	[cosθ 0 sinθ; 0 1 0; −sinθ 0 cosθ]	y-coordinate unchanged	xz-plane
z-axis: R_z(θ)	[cosθ −sinθ 0; sinθ cosθ 0; 0 0 1]	z-coordinate unchanged	xy-plane

Reflections in R²

Reflection across the

x

-axis flips the

y

-coordinate:

T(x, y) = (x, -y)

, with matrix

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

.

Reflection across the

y

-axis flips the

x

-coordinate: matrix

\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

.

Reflection across the line

y = x

swaps coordinates: matrix

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

.

Reflection across an arbitrary line through the origin at angle

\alpha

has matrix

Reflection Across Line 2D

H_\alpha = \begin{pmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{pmatrix}

Learn more about this formula: Reflection Across Line 2D →

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All reflection matrices share the same properties: the determinant is

-1

(orientation-reversing, area-preserving), the matrix is orthogonal (lengths and angles preserved), and the matrix is involutory (

H^2 = I

) — reflecting twice returns every vector to its starting point. The eigenvalues are

+1

(vectors on the mirror line) and

-1

(vectors perpendicular to it).

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Transformations · reflections

Reflections in the plane

Four mirrors, four matrices. The first three are special cases of the fourth — substituting $\alpha = 0$ , $\tfrac{\pi}{2}$ and $\tfrac{\pi}{4}$ into the general form recovers them exactly.

4mirrors

Mirrors along the axes2

The x-axis§ 5

\alpha = 0

matrix

\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}

effect on (x, y)

(x, y) \mapsto (x, -y)

fixed directionthe

x

-axis

Diagonal, so its action is entrywise: the first coordinate survives and the second flips. The eigenvalues sit on the diagonal,

+1

and

-1

, exactly as the general result predicts.

The y-axis§ 5

\alpha = \tfrac{\pi}{2}

matrix

\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}

effect on (x, y)

(x, y) \mapsto (-x, y)

fixed directionthe

y

-axis

The same matrix with the signs exchanged. Note this is not the negative of the row above — that would be

-I

, a rotation by

\pi

, which reverses both coordinates and has

\det = +1

.

Mirror at 45 degrees1

The line

y = x

§ 5

\alpha = \tfrac{\pi}{4}

matrix

\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

effect on (x, y)

(x, y) \mapsto (y, x)

fixed directionthe line

y = x

Swapping coordinates is a reflection, which is worth noticing because it also describes the transpose at the level of a

2 \times 2

permutation. Its own inverse, as every reflection is.

The general mirror1

Line at angle

\alpha

§ 5any line through the origin

matrix

\begin{bmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{bmatrix}

effect on (x, y)reflects across that line

fixed directionthe mirror line itself

The

2\alpha

is not a typo: rotating the mirror by

\alpha

turns the image by

2\alpha

, since the incoming and outgoing rays each make angle

\alpha

with it. Symmetric and orthogonal at once, which is unusual and is why

A^{-1} = A^{\mathsf{T}} = A

.

Every reflection has

\det = -1

, eigenvalues

+1

along the mirror and

-1

across it, and equals its own inverse. Those three facts hold whatever

\alpha

is, which is what makes reflection a family rather than a list — and why the general row is the only one that has to be remembered.

Reflections in the plane·/linear-algebra/transformations/geometricLearn Math Class

The doubled angle in the general matrix is the detail worth pausing on. Rotating the mirror by

\alpha

moves the image by

2\alpha

, because the incoming ray and the outgoing ray each make angle

\alpha

with the line — so the total turn is twice the tilt. It also explains why composing two reflections gives a rotation by twice the angle between their mirrors, which is the fastest route to that result.

Reflection in an axis, then in a diagonal

Points on the mirror line stay exactly where they are while everything else swaps to the far side at equal distance. Changing the line changes the matrix but not the behaviour, and applying either matrix twice returns every point to its start. Rotate the mirror line and watch the entries change on the reflection explorer.

Any matrix that is its own inverse and has determinant −1 in two dimensions is a reflection of this kind.

Reflections in R³

Reflection across a coordinate plane negates the coordinate perpendicular to that plane. Reflection across the

xy

-plane:

\text{diag}(1, 1, -1)

. Across the

xz

-plane:

\text{diag}(1, -1, 1)

. Across the

yz

-plane:

\text{diag}(-1, 1, 1)

.

Reflection across an arbitrary plane through the origin with unit normal

\mathbf{n}

is given by the Householder matrix:

Householder Reflection

H = I - 2\,\mathbf{n}\mathbf{n}^T

Learn more about this formula: Householder Reflection →

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This matrix subtracts twice the component of each vector in the direction of

\mathbf{n}

, effectively mirroring across the plane perpendicular to

\mathbf{n}

. Householder reflections are orthogonal, have determinant

-1

, and satisfy

H^2 = I

. They are the building blocks of the QR decomposition and are widely used in numerical algorithms.

Projections

Orthogonal projection onto a subspace collapses each vector onto its nearest point in the subspace, discarding the perpendicular component.

Projection onto the

x

-axis in

\mathbb{R}^2

:

T(x, y) = (x, 0)

, with matrix

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}

.

Projection onto a line through the origin in direction

\mathbf{u}

:

Projection onto Line

P = \frac{\mathbf{u}\mathbf{u}^T}{\mathbf{u}^T\mathbf{u}}

Learn more about this formula: Projection onto Line →

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Projection onto a plane with unit normal

\mathbf{n}

in

\mathbb{R}^3

:

Projection onto Plane

P = I - \frac{\mathbf{n}\mathbf{n}^T}{\mathbf{n}^T\mathbf{n}}

Learn more about this formula: Projection onto Plane →

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All orthogonal projection matrices share the same algebraic signature:

P^2 = P

(idempotent — projecting twice is the same as projecting once),

P^T = P

(symmetric), eigenvalues are

0

and

1

, and

\text{rank}(P) = \text{tr}(P)

. The determinant is

0

unless the projection is onto the full space — projections always collapse at least one dimension.

The plane dropped onto a line

Unlike a reflection, this one destroys information: two different points can land on the same place and there is no way back. The determinant is zero and the matrix is idempotent, which are the algebraic signatures of exactly that loss. Compare it against the reflection above on the 2D projection explorer.

Reflections and rotations preserve information; projections and their relatives are where it starts being lost.

Shears

A shear displaces each point in proportion to its distance from a fixed line or plane. The two

2 \times 2

shear matrices are:

Shear Matrix

\text{Shear}_x = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}, \qquad \text{Shear}_y = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}

Learn more about this formula: Shear Matrix →

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In

\mathbb{R}^2

, a horizontal shear shifts the

x

-coordinate by

k

times the

y

-coordinate:

T(x, y) = (x + ky, \; y), \quad \text{matrix } \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

A vertical shear shifts the

y

-coordinate by

k

times the

x

-coordinate:

T(x, y) = (x, \; kx + y), \quad \text{matrix } \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}

Both are triangular matrices with determinant

1

: shears are area-preserving and orientation-preserving. They are not orthogonal — angles are distorted. A square sheared horizontally becomes a parallelogram of the same area but with tilted sides.

In

\mathbb{R}^3

, there are six possible shear directions (each coordinate shifted by a multiple of each other coordinate). Each is represented by an identity matrix with one off-diagonal entry replaced by

k

. Shears are fundamental building blocks — any invertible matrix with determinant

1

can be written as a product of shears.

Combining Transformations

Composing transformations corresponds to multiplying their matrices. A rotation followed by a scaling is

SA

where

S

is the scaling matrix and

A

is the rotation matrix. The product is applied right-to-left:

A

acts first, then

S

.

Order matters. Rotating then shearing produces a different result from shearing then rotating:

\text{Shear} \cdot \text{Rotation} \neq \text{Rotation} \cdot \text{Shear}

in general.

The singular value decomposition reveals the hidden geometric structure of any matrix:

A = U\Sigma V^T

, where

V^T

is a rotation (or rotation-reflection),

\Sigma

is a coordinate-axis scaling, and

U

is another rotation (or rotation-reflection). Every linear transformation is a rotation, followed by a scaling along the coordinate axes, followed by another rotation. The singular values in

\Sigma

measure the maximum stretching in each orthogonal direction.

This decomposition means no linear transformation is truly exotic — even the most complex-looking matrix is just three simple geometric operations composed together.

Determinant as Geometric Signature

The determinant classifies every linear transformation by its effect on size and orientation.

|\det(A)|

is the factor by which the transformation scales areas (in

\mathbb{R}^2

) or volumes (in

\mathbb{R}^3

). A unit square maps to a parallelogram of area

|\det(A)|

. A unit cube maps to a parallelepiped of volume

|\det(A)|

.

\det(A) > 0

: the transformation preserves orientation. Counterclockwise stays counterclockwise in

\mathbb{R}^2

; right-handed stays right-handed in

\mathbb{R}^3

. Rotations and shears fall in this category.

\det(A) < 0

: the transformation reverses orientation. Counterclockwise becomes clockwise; right-handed becomes left-handed. Reflections are the canonical example.

\det(A) = 0

: the transformation collapses at least one dimension. The image is a proper subspace — a line or point in

\mathbb{R}^2

, a plane, line, or point in

\mathbb{R}^3

. Projections onto proper subspaces and singular matrices fall here.

|\det(A)| = 1

: the transformation preserves area or volume. Rotations (

\det = +1

) and reflections (

\det = -1

) are the area-preserving and volume-preserving transformations. Shears also have

\det = 1

, preserving area despite distorting angles.

det(A)	Effect on area / volume	Effect on orientation	Typical transformations
\|det(A)\| > 1	expands by factor \|det(A)\|	preserves if det > 0, reverses if det < 0	uniform scaling with \|c\| > 1
0 < \|det(A)\| < 1	shrinks by factor \|det(A)\|	preserves if det > 0, reverses if det < 0	uniform scaling with 0 < \|c\| < 1
det(A) = +1	preserves area / volume	preserves orientation	rotation, shear
det(A) = −1	preserves area / volume	reverses orientation (handedness flip)	reflection, improper rotation
det(A) = 0	collapses at least one dimension (image is a proper subspace)	undefined / degenerate	projection onto a proper subspace, any singular matrix

Area scaled, then area destroyed

The determinant reports what the transformation does to area: above, the unit square has become a parallelogram of some definite size; below, it has been flattened to nothing. A negative value would mean the square had been turned over as well. One number, carrying both the scaling and the orientation. Change the matrix and watch it respond on the linear transformation explorer.

This is why the determinant can classify a transformation family before any of its entries are examined individually.

Summary: Algebraic Signature of Each Family

Each family of geometric transformations has been treated in its own section, with its own matrix templates and its own algebraic identities. The table below sets all five side by side across the diagnostic signatures that distinguish them — orthogonality, determinant, eigenvalues, the characteristic algebraic identity, and what each family does to length and to area. It is the recognition card to keep nearby when a matrix is in hand and the question is which geometric family it belongs to.

Read the other way round, this is a recognition problem: given a matrix and no picture, which family is it? The two columns below answer that without any geometry — an identity the matrix satisfies, and the spectrum it must have. Both are checkable by computation, which matters because in dimensions above three the picture is unavailable anyway.

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Transformations · signatures

Recognising a family from its matrix

Each family has an algebraic fingerprint — an identity its matrix satisfies and a spectrum it must have. Given an unfamiliar matrix, these are what identify it without drawing anything.

5families

Rigid — lengths preserved2

1

Rotation§ 3

eigenvalues

e^{\pm i\theta}

— complex pair

R^{\mathsf{T}}R = I, \quad \det R = 1

Orthogonal with positive determinant. Over

\mathbb{R}

a plane rotation has no eigenvectors at all, since no direction survives unturned — which is exactly what the complex pair records. In

\mathbb{R}^3

there is one real eigenvalue

1

, and its eigenvector is the axis.

2

Reflection§ 5

eigenvalues

+1

on the mirror,

-1

across it

H^2 = I, \quad H^{\mathsf{T}} = H, \quad \det H = -1

Involutory and symmetric. Applying it twice returns the original, so

H

is its own inverse. The eigenvalue

-1

is what the negative determinant reports — orientation reverses, and no rotation can do that.

Non-rigid — lengths change2

3

Uniform scaling§ 2

eigenvalue

c

, repeated

n

times

A = cI, \quad \det A = c^n

The only family commuting with everything, since

cI

passes through any product. Every direction is an eigenvector, so the matrix is diagonal in every basis — the extreme opposite of a shear.

4

Shear§ 8

eigenvalue

1

repeated — defective

triangular,

1

s on the diagonal,

\det = 1

Area is preserved yet nothing is rigid. The repeated eigenvalue has only one eigenvector rather than two, so a shear cannot be diagonalised — the standard example of algebraic multiplicity exceeding geometric.

Not invertible1

5

Orthogonal projection§ 7

eigenvalues

0

and

1

only

P^2 = P, \quad P^{\mathsf{T}} = P, \quad \det P = 0

Idempotent and symmetric. The zero determinant is not incidental — a projection collapses the perpendicular directions and nothing recovers them, so it is the only family here with no inverse. See projection properties.

The determinant sorts them at a glance: rotations and shears preserve area, reflections flip it, projections destroy it, and scaling multiplies it by

c^n

. Two families share

\det = 1

— rotation and shear — and the eigenvalues separate them: a rotation has a complex pair, a shear has a repeated real one and too few eigenvectors to diagonalise.

Recognising a family from its matrix·/linear-algebra/transformations/geometricLearn Math Class

Notice which pairs are hard to tell apart. Rotation and shear both have

\det = 1

, so area alone will not separate them — the eigenvalues do, a complex pair against a defective repeated root. Reflection and projection are both symmetric, and there the determinant separates them:

-1

against

0

, orientation reversed against information destroyed. No single test identifies a family; the identity and the spectrum together do.

Geometric Transformations FAQ

In $R_{\theta}S_k$ , does the rotation or the scaling happen first?

+

The scaling, because the rightmost factor acts first. Reading the product left to right as "rotate, then scale" reverses the actual sequence, and the mistake stays invisible until the two operations fail to commute. Here

AB \neq BA

carries real geometric weight: rotating then reflecting is a different motion from reflecting then rotating.Read more →

How do you invert a rotation matrix?

+

You do not need to compute anything:

R_{\theta}^{-1} = R_{-\theta} = R_{\theta}^{T}

. Because a rotation is orthogonal, its transpose already is its inverse, so running elimination on one is wasted effort. Negating the angle and transposing the matrix are two descriptions of the same operation.Read more →

Why does $P$ mean different things in different places?

+

It has three unrelated tenants in linear algebra: a projection matrix here, a change-of-basis matrix in coordinate work, and a permutation matrix in factorisations such as

PA = LU

. Only the subscript and the surrounding claim distinguish them, so the letter alone never settles which object is meant.Read more →

Geometric Transformations

Rotations, Reflections, Projections, and More

How to Read a Transformation Matrix

Geometric Transformation Notation

Geometric Transformation Notation

Scaling

Rotations in R²

Rotations in R³

Reflections in R²

Reflections in the plane

Reflections in R³

Projections

Shears

Combining Transformations

Determinant as Geometric Signature

Summary: Algebraic Signature of Each Family

Recognising a family from its matrix

Geometric Transformations FAQ

In RθSkR_{\theta}S_kRθ​Sk​, does the rotation or the scaling happen first?

How do you invert a rotation matrix?

Why does PPP mean different things in different places?

In $R_{\theta}S_k$ , does the rotation or the scaling happen first?

Why does $P$ mean different things in different places?