MathematicsLinear Algebra and TransformationsA-Level

Rotation Matrix (2D)

This matrix performs a counter-clockwise rotation of a point or vector in 2D Cartesian space by an angle θ about the origin.

Understand the formulaSee the free derivationOpen the full walkthrough

R_{θ} = (cos θ - sin θ sin θ cos θ)

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Core idea

Overview

The matrix operates on a column vector by multiplying it to transform coordinates while preserving the origin's position and the lengths of vectors. It is a fundamental component of linear transformations, illustrating how geometric operations are expressed as algebraic matrix operations.

When to use: Apply this when you need to find the new coordinates of a point after it has been rotated by a specific angle around the origin.

Why it matters: It is essential in computer graphics, game engine development, and robotics for calculating the orientation and movement of objects in 2D space.

Symbols

Variables

$θ$ = Rotation Angle, x = Initial X-coordinate, y = Initial Y-coordinate

θ

Rotation Angle

Variable

x

Initial X-coordinate

Variable

y

Initial Y-coordinate

Variable

Walkthrough

Derivation

Derivation of Rotation Matrix (2D)

The 2D rotation matrix is derived by observing the image of the standard basis vectors under a rotation of θ about the origin.

The transformation is linear, meaning it can be represented by a matrix.
Rotation is performed in the counter-clockwise direction in a 2D Cartesian plane.

Define basis vectors

Identify the standard basis vectors for 2D space, which correspond to the columns of the transformation matrix.

i = (10), j = (01)

Note: A matrix M is formed by [M(i) M(j)].

Rotate the basis vectors

Applying trigonometry, rotating (1,0) by θ results in (cos θ, sin θ). Rotating (0,1) by θ results in (cos(θ+90°), sin(θ+90°)).

i^{'} = (cos θ sin θ), j^{'} = (cos (θ + 9 0^{\circ}) sin (θ + 9 0^{\circ}))

Note: Remember that rotating 90° counter-clockwise maps (1,0) to (0,1) and (0,1) to (-1,0).

Apply trigonometric identities

Simplify the coordinates of the rotated basis vector j' using standard trigonometric co-function identities.

cos (θ + 9 0^{\circ}) = - sin θ, sin (θ + 9 0^{\circ}) = cos θ

Note: Use the unit circle to visualize these sign changes.

Construct the matrix

Assemble the transformed basis vectors as the columns of the rotation matrix to obtain the final formula.

R_{θ} = (i^{'} j^{'}) = (cos θ - sin θ sin θ cos θ)

Note: Ensure the order of columns is preserved to maintain the correct transformation.

Result

R_{θ} = (i^{'} j^{'}) = (cos θ - sin θ sin θ cos θ)

Source: A-Level Mathematics Specification: Further Pure Mathematics (Matrices and Transformations)

Free formulas

Rearrangements

Solve for $θ$

Make $θ$ the subject

θ = arccos (R_{11})

Isolate the angle $θ$ by using inverse trigonometric functions on the elements of the rotation matrix.

Difficulty: 4/5

Solve for $R_{11}$

Make $R_{11}$ the subject

R_{11} = cos θ

Extract the value directly from the top-left element of the matrix.

Difficulty: 1/5

Solve for $R_{12}$

Make $R_{12}$ the subject

R_{12} = - sin θ

Extract the value directly from the top-right element of the matrix.

Difficulty: 1/5

Solve for $R_{21}$

Make $R_{21}$ the subject

R_{21} = sin θ

Extract the value directly from the bottom-left element of the matrix.

Difficulty: 1/5

Solve for $R_{22}$

Make $R_{22}$ the subject

R_{22} = cos θ

Extract the value directly from the bottom-right element of the matrix.

Difficulty: 1/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Why it behaves this way

Intuition

Imagine the unit basis vectors î (1,0) and ĵ (0,1) as two stiff rods connected at the origin. When you rotate the entire coordinate system counter-clockwise by θ, the vector î lands at (cos θ, sin θ), and the perpendicular vector ĵ lands at (-sin θ, cos θ). The matrix is simply the coordinate representation of these two new rotated positions placed side-by-side as columns.

θ

Angle of rotation

The amount of 'swing' applied to the plane, measured in radians or degrees.

cos θ

Horizontal component

The 'shadow' cast on the x-axis by a rotated unit vector.

s in θ

Vertical component

The 'height' or 'shadow' cast on the y-axis by a rotated unit vector.

Signs and relationships

-sinθ: When rotating counter-clockwise, the x-component of the original ĵ vector (which starts at 90 degrees) must push into the negative x-direction as it moves toward the second quadrant.

One free problem

Practice Problem

Rotate the point (1, 0) counter-clockwise by 90 degrees. What is the new x-coordinate?

Rotation Angle90

Initial X-coordinate1

Initial Y-coordinate0

Solve for: $y_{n} e w$

Hint: Use cos(90) = 0 and sin(90) = 1 in the rotation matrix formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A character in a 2D side-scrolling video game rotates to face a new direction based on input from the player's controller.

Study smarter

Tips

Remember that a negative angle corresponds to a clockwise rotation.
Ensure your calculator is in the correct mode (degrees or radians) before calculating the trig values.
The matrix is orthogonal, meaning its inverse is simply its transpose, which corresponds to rotating by -θ.

Avoid these traps

Common Mistakes

Swapping the positions of sine and cosine components.
Incorrectly placing the negative sign on the sine in the wrong row or column.
Mixing up degrees and radians when performing calculations.

Common questions

Frequently Asked Questions

The 2D rotation matrix is derived by observing the image of the standard basis vectors under a rotation of θ about the origin.

Apply this when you need to find the new coordinates of a point after it has been rotated by a specific angle around the origin.

It is essential in computer graphics, game engine development, and robotics for calculating the orientation and movement of objects in 2D space.

Swapping the positions of sine and cosine components. Incorrectly placing the negative sign on the sine in the wrong row or column. Mixing up degrees and radians when performing calculations.

A character in a 2D side-scrolling video game rotates to face a new direction based on input from the player's controller.

Remember that a negative angle corresponds to a clockwise rotation. Ensure your calculator is in the correct mode (degrees or radians) before calculating the trig values. The matrix is orthogonal, meaning its inverse is simply its transpose, which corresponds to rotating by -θ.

References

Sources

Stewart, J. (2015). Calculus: Early Transcendentals.
Strang, G. (2016). Introduction to Linear Algebra.
A-Level Mathematics Specification: Further Pure Mathematics (Matrices and Transformations)

Rotation Matrix (2D)

Overview

Variables

Derivation

Define basis vectors

Rotate the basis vectors

Apply trigonometric identities

Construct the matrix

Rearrangements

Graph

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Determinant of a Matrix

Matrix Multiplication

Frequently Asked Questions

Sources