Multiplying the rotation matrix by the point vector produces:

(x, y) → (−y, x)

Common Transformation Matrices

Each pair in the table represents one row of a 2 × 2 matrix.

General Rotation Matrix

For a counterclockwise rotation through angle θ:

R = [[cos θ, −sin θ], [sin θ, cos θ]]

Applying the matrix gives:

x′ = x cos θ − y sin θ

y′ = x sin θ + y cos θ

Worked Example: Rotation Through 30°

Rotate:

P = (4, 0)

30° counterclockwise about the origin.

Step 1: Apply the formulas

x′ = 4 cos 30° − 0 sin 30°

y′ = 4 sin 30° + 0 cos 30°

Step 2: Evaluate

x′ = 4(√3/2) = 2√3

y′ = 4(1/2) = 2

Final Answer

P′ = (2√3, 2)

Approximately:

P′ ≈ (3.46, 2)

Why Translations Are Different from Linear Matrix Transformations

A translation adds a vector to a point:

P′ = P + v

A rotation or reflection about the origin can be represented using a 2 × 2 matrix:

P′ = MP

A nonzero translation cannot be represented by multiplying ordinary two-dimensional position vectors by a 2 × 2 matrix alone.

This is because every linear transformation sends the origin to the origin, while a nonzero translation moves it.

However, translations and linear transformations can be combined using:

P′ = MP + v

This is called an affine transformation.

Example

A point is rotated 90° counterclockwise and then translated by (3, 2).

The transformation is:

P′ = RP + (3, 2)

For:

P = (2, 1)

First rotate:

RP = (−1, 2)

Then translate:

P′ = (−1, 2) + (3, 2)

P′ = (2, 4)

This mathematical form is widely used in computer graphics and robotics.

Homogeneous Coordinates

In computer graphics, translations can be included in matrix multiplication by using homogeneous coordinates.

A two-dimensional point:

P = (x, y)

is represented as:

P = (x, y, 1)

A translation by (a, b) can then be represented using the matrix:

T = [[1, 0, a], [0, 1, b], [0, 0, 1]]

Multiplying T by the homogeneous point vector gives:

(x, y, 1) → (x + a, y + b, 1)

Why This Matters

Homogeneous coordinates allow translations, rotations, reflections, and dilations to be combined into matrix operations.

This makes complex transformation sequences easier to calculate and implement in computer systems.

Vector Transformations in Three Dimensions

Translations can also be performed in three-dimensional space.

For a point:

P = (x, y, z)

and translation vector:

v = (a, b, c)

the transformation is:

(x, y, z) → (x + a, y + b, z + c)

Worked Example

A point has coordinates:

P = (2, −1, 4)

Translate it by:

v = (3, 5, −2)

Step 1: Add corresponding components

P′ = (2 + 3, −1 + 5, 4 − 2)

Final Answer

P′ = (5, 4, 2)

Finding the Magnitude of a Three-Dimensional Translation

The magnitude of:

v = (a, b, c)

is:

|v| = √(a² + b² + c²)

For:

v = (3, 5, −2)

|v| = √(3² + 5² + (−2)²)

|v| = √(9 + 25 + 4)

|v| = √38 ≈ 6.16 units

Transformations of Lines and Planes

Vectors can describe transformations of entire geometric objects, not just individual points.

Translating a Line

Suppose a line has vector equation:

r = a + td

Translate every point by vector v.

The transformed line is:

r′ = (a + v) + td

The direction vector remains unchanged.

Worked Example

Original line:

r = (1, 2, 3) + t(2, 1, −1)

Translation:

v = (4, −2, 5)

Step 1: Translate the known point

a′ = (1, 2, 3) + (4, −2, 5)

a′ = (5, 0, 8)

Step 2: Keep the direction vector

d = (2, 1, −1)

Final Answer

r′ = (5, 0, 8) + t(2, 1, −1)

The translated line is parallel to the original line.

Translating a Plane

Consider a plane:

n · r = d

where n is the plane's normal vector.

After translation by v, its equation becomes:

n · r′ = d + n · v

The normal vector does not change.

Worked Example

Original plane:

2x + y − z = 5

Translation:

v = (3, 1, 2)

The normal vector is:

n = (2, 1, −1)

Calculate:

n · v = (2)(3) + (1)(1) + (−1)(2)

n · v = 6 + 1 − 2

n · v = 5

Therefore, the translated plane is:

2x + y − z = 10

The plane's orientation remains unchanged.

Analysing Geometric Movement

Vector transformations allow us to compare displacement, distance travelled, and final position.

These quantities are related but are not identical.

Displacement

Displacement describes the change in position from start to finish.

Displacement = final position − initial position

Distance Travelled

Distance travelled is the total length of the actual path.

For several straight-line movements:

Total distance = |v₁| + |v₂| + ...

Worked Example

A person walks:

• 6 m east.

• 8 m north.

Represent the movements as:

v₁ = (6, 0)

v₂ = (0, 8)

Step 1: Calculate the displacement

v_total = (6, 0) + (0, 8)

v_total = (6, 8)

Step 2: Calculate displacement magnitude

|v_total| = √(6² + 8²)

|v_total| = 10 m

Step 3: Calculate distance travelled

Distance = 6 + 8

Distance = 14 m

Final Answers

Displacement = 10 m northeast

Distance travelled = 14 m

The distance travelled is greater because the actual path is not a straight line between the starting and ending positions.

Real-World Application: Navigation

Navigation systems use vectors to represent movement and calculate final positions.

A vehicle or aircraft may undergo a series of displacements.

The overall displacement is found by adding the movement vectors.

Worked Example

A drone begins at:

P = (10, 20, 5)

Its movements are:

v₁ = (4, 3, 2)

v₂ = (−1, 5, −1)

v₃ = (2, −4, 3)

Find the final position.

Step 1: Calculate total displacement

v_total = (4, 3, 2) + (−1, 5, −1) + (2, −4, 3)

Step 2: Combine components

x-component:

4 − 1 + 2 = 5

y-component:

3 + 5 − 4 = 4

z-component:

2 − 1 + 3 = 4

Therefore:

v_total = (5, 4, 4)

Step 3: Calculate final position

P′ = (10, 20, 5) + (5, 4, 4)

Final Answer

P′ = (15, 24, 9)

The drone finishes at coordinates (15, 24, 9).

Real-World Application: Computer Graphics and Animation

Computer graphics systems use transformations to move, rotate, resize, and orient objects.

For example, a video game character may:

• Translate across a scene.

• Rotate to face a target.

• Scale during an animation.

• Reflect in a mirror.

• Move through a three-dimensional environment.

Why Vector Transformations Are Useful

A three-dimensional object may contain thousands of vertices.

Instead of manually recalculating each vertex, software applies the same transformation rule to every vertex.

For example:

P′ = MP + v

can represent a rotation, scaling operation, or reflection followed by a translation.

Complex animations can be produced by combining many such transformations.

Real-World Application: Robotics

Robotic arms move through a sequence of rotations and translations.

Each joint changes the position and orientation of the components attached to it.

Example

A robotic tool begins at:

P = (2, 3)

It undergoes:

1. A translation of (4, 0).

2. A 90° counterclockwise rotation about the origin.

3. A translation of (0, −2).

Find the final position.

Step 1: First translation

P′ = (2 + 4, 3 + 0)

P′ = (6, 3)

Step 2: Rotation

(6, 3) → (−3, 6)

Step 3: Second translation

P‴ = (−3, 6) + (0, −2)

Final Answer

P‴ = (−3, 4)

The order of operations is essential.

Real-World Application: Physics

In physics, vector transformations can describe changes in position and coordinate orientation.

For example:

• Displacement vectors describe movement.

• Rotation matrices describe changes in direction.

• Coordinate transformations allow the same motion to be analysed from different reference frames.

• Three-dimensional transformations help describe rigid-body motion.

Example: Changing Coordinate Orientation

Suppose a velocity vector is:

v = (5, 0)

If the coordinate axes are rotated 90° counterclockwise while the physical velocity remains unchanged, the velocity's components in the new coordinate system become:

v_new = (0, −5)

This is different from actively rotating the velocity vector itself 90° counterclockwise in a fixed coordinate system, which would give:

v_rotated = (0, 5)

Important Distinction

An active transformation changes the geometric object or vector.

A passive transformation changes the coordinate system used to describe it.

This distinction becomes important in advanced physics, engineering, and robotics.

Multi-Step Transformation Problems

Worked Example: Translation, Reflection, and Rotation

A point begins at:

P = (3, 1)

It undergoes:

1. Translation by (−2, 4).

2. Reflection across the y-axis.

3. Rotation by 90° clockwise about the origin.

Find the final coordinates.

Step 1: Translate

P′ = (3 − 2, 1 + 4)

P′ = (1, 5)

Step 2: Reflect Across the y-Axis

Rule:

(x, y) → (−x, y)

Therefore:

P″ = (−1, 5)

Step 3: Rotate 90° Clockwise

Rule:

(x, y) → (y, −x)

Therefore:

P‴ = (5, 1)

Final Answer

P‴ = (5, 1)

Verification

Each transformation has been applied in the stated order.

Reversing the order could produce a different final point.

Worked Example: Transforming a Triangle

A triangle has vertices:

A = (1, 1)

B = (4, 1)

C = (2, 3)

The triangle is:

1. Translated by (2, 1).

2. Rotated 180° about the origin.

Find the final coordinates.

Step 1: Translate Each Vertex

A′ = (3, 2)

B′ = (6, 2)

C′ = (4, 4)

Step 2: Apply the 180° Rotation

Rule:

(x, y) → (−x, −y)

Therefore:

A″ = (−3, −2)

B″ = (−6, −2)

C″ = (−4, −4)

Final Answer

The transformed triangle has vertices:

A″ = (−3, −2)

B″ = (−6, −2)

C″ = (−4, −4)

Geometric Interpretation

Both translations and rotations are rigid transformations.

Therefore, the final triangle is congruent to the original triangle.

Its side lengths, angles, perimeter, and area remain unchanged.

Analysing Transformation Properties

Different transformations preserve different geometric properties.

Orientation

Orientation describes the ordering of points around a figure.

For example, a triangle's vertices may be arranged clockwise or counterclockwise.

Translations and rotations preserve this ordering.

Reflections reverse it.

A dilation with a negative scale factor centred at a point corresponds to a positive dilation combined with a 180° rotation, so it preserves orientation in two dimensions.

Choosing the Correct Transformation Method

General Problem-Solving Strategy

1. Identify the original point or figure.

2. Determine the type of transformation.

3. Write the transformation rule.

4. Apply the rule to each relevant point.

5. If several transformations are involved, follow the stated order.

6. Calculate the final coordinates.

7. Check whether the expected geometric properties are preserved.

8. Interpret the movement in the original context.

9. Verify the result using an inverse transformation or coordinate substitution when appropriate.

Common Mistakes

Confusing a point with a translation vector

A point represents a location, while a translation vector represents movement.

Subtracting instead of adding during translation

Translations are performed by adding the translation vector to the point coordinates.

Ignoring negative components

Negative components indicate movement left or downward in the standard coordinate system.

Translating only one vertex of a figure

Every vertex must undergo the same translation.

Confusing distance travelled with displacement

Displacement is the net change in position, while distance travelled measures the total path length.

Adding vector magnitudes instead of vector components

The combined translation is found by adding corresponding components.

Using the wrong reflection rule

Different reflection lines require different coordinate rules.

Confusing clockwise and counterclockwise rotations

A 90° clockwise rotation differs from a 90° counterclockwise rotation.

Forgetting the centre of rotation

Rotation rules depend on the centre of rotation.

Assuming transformation order never matters

Pure translations commute, but combinations of different transformations generally do not.

Confusing congruence and similarity

Rigid transformations preserve size and shape, while dilations preserve shape but generally change size.

Using an incorrect scale factor

A dilation by k multiplies coordinates relative to its centre by k.

Assuming a 2 × 2 matrix can represent every translation

Nonzero translations require vector addition or an augmented matrix representation.

Applying matrix transformations in the wrong order

Matrix multiplication is generally not commutative.

Confusing active and passive transformations

Moving a vector and changing the coordinate system are different operations.

Failing to check geometric properties

Translations and rotations should preserve lengths and angles.

Rounding intermediate calculations too early

Keep exact values whenever possible and round only the final result.

Key Terms

• Vector: A quantity with magnitude and direction.

• Position Vector: A vector describing the location of a point relative to the origin.

• Translation: A transformation that moves every point by the same displacement.

• Translation Vector: A vector describing the distance and direction of a translation.

• Geometric Transformation: A rule mapping points of a figure to new positions.

• Preimage: The original figure before a transformation.

• Image: The figure produced by a transformation.

• Coordinate Rule: A mathematical rule describing how coordinates change.

• Rigid Transformation: A transformation preserving distances and angles.

• Isometry: Another name for a distance-preserving transformation.

• Reflection: A transformation that flips a figure across a line.

• Line of Reflection: The line across which a figure is reflected.

• Rotation: A transformation that turns a figure around a fixed point.

• Centre of Rotation: The fixed point around which a rotation occurs.

• Angle of Rotation: The angle through which a figure is rotated.

• Dilation: A transformation that changes the size of a figure using a scale factor.

• Scale Factor: A number determining the enlargement or reduction of a figure.

• Composition of Transformations: A sequence of two or more transformations.

• Inverse Transformation: A transformation that reverses another transformation.

• Identity Transformation: A transformation that leaves every point unchanged.

• Displacement: The change in position from an initial point to a final point.

• Distance Travelled: The total length of the path followed.

• Magnitude: The length or size of a vector.

• Direction Angle: An angle describing the direction of a vector relative to a reference axis.

• Matrix: A rectangular arrangement of numbers used in mathematical operations.

• Transformation Matrix: A matrix representing a geometric transformation.

• Linear Transformation: A transformation preserving vector addition and scalar multiplication.

• Affine Transformation: A transformation combining a linear transformation with a translation.

• Homogeneous Coordinates: An extended coordinate system allowing translations to be represented using matrices.

• Commutative Property: A property stating that changing the order of operations does not change the result.

• Congruence: A relationship between figures with the same shape and size.

• Similarity: A relationship between figures with the same shape and proportional dimensions.

• Orientation: The clockwise or counterclockwise ordering of points in a figure.

• Active Transformation: A transformation that changes the position or orientation of an object.

• Passive Transformation: A change in the coordinate system used to describe an object.

• Three-Dimensional Translation: A movement described by changes in x-, y-, and z-coordinates.

• Geometric Modelling: Using mathematical representations to describe shapes, positions, and movement.

Key Takeaways

• Vector transformations provide a mathematical method for describing geometric movement.

• A translation moves every point of a figure by the same vector.

• In two dimensions, a translation by (a, b) follows the rule (x, y) → (x + a, y + b).

• In three dimensions, a translation by (a, b, c) follows the rule (x, y, z) → (x + a, y + b, z + c).

• A translation vector can be found by subtracting the original coordinates from the transformed coordinates.

• The magnitude of a translation vector gives the straight-line displacement distance.

• Multiple translations can be combined by adding their vectors.

• The order of pure translations does not affect the final result.

• An inverse translation uses the negative of the original translation vector.

• Reflections reverse orientation while preserving distances and angles.

• Rotations preserve distances, angles, and orientation.

• Dilations change size according to a scale factor.

• Different transformation types can be combined into a composition.

• The order of transformations generally matters.

• Matrices provide a systematic way to represent rotations, reflections, and dilations.

• Affine transformations combine matrix operations with vector translations.

• Homogeneous coordinates allow translations to be represented through matrix multiplication.

• Translations can be applied to points, figures, lines, and planes.

• Vector addition allows us to analyse total displacement from several movements.

• Displacement and distance travelled are different quantities.

• Vector transformations are widely used in navigation, robotics, physics, animation, and computer graphics.

• The central skill is applying vector rules to describe, combine, analyse, and model geometric transformations in two and three dimensions.

Check Your Understanding

Recall and Understanding

1. Define a translation and explain how a vector describes its movement.

2. Explain the difference between a preimage and an image.

3. State the coordinate rule for a translation by vector (a, b).

4. Explain why the order of two translations does not affect the final result.

5. Describe the difference between a rigid transformation and a dilation.

Application

6. Translate P = (3, −2) by vector v = (5, 4). Find P′.

7. A triangle has vertices A = (1, 2), B = (4, 2), and C = (2, 5). Translate it by (−3, 1).

8. Point P = (−4, 6) is translated to P′ = (2, 1). Find the translation vector.

9. Combine the translations v₁ = (4, 3), v₂ = (−2, 5), and v₃ = (1, −4).

10. Rotate P = (5, 2) by 90° counterclockwise about the origin.

Analysis and Challenge

11. Point P = (2, 3) is translated by (4, −1), then reflected across the y-axis. Find its final coordinates.

12. Point P = (3, −1) is rotated 180° about the origin and then translated by (2, 5). Find its final coordinates.

13. A robot begins at (5, 2, 1) and moves by vectors (3, 4, 2), (−2, 1, −3), and (4, −5, 1). Find its final position and the magnitude of its total displacement.

14. A line has equation r = (1, 2, 3) + t(2, −1, 4). Translate it by vector (3, 5, −2). Write the new line equation.

15. A plane has equation 2x − y + 3z = 7. Translate it by vector (1, 2, −1). Determine the equation of the translated plane.

Final Summary

Vector transformations provide a systematic mathematical framework for describing how geometric objects move and change in space.

Translations are represented by adding displacement vectors to points. Multiple translations can be combined through vector addition, and inverse translations reverse the original movement.

Reflections, rotations, and dilations can be described using coordinate rules and matrices.

More complex transformations can be modelled using affine transformations, which combine linear transformations with translations.

The order of operations is important when several different transformations are applied.

Vector transformations are useful for describing geometric movement, analysing displacement, modelling rigid-body motion, and performing calculations in computer graphics, engineering, robotics, and navigation.

The central principle is that geometric transformations can be represented using mathematical operations on vectors and coordinates, allowing us to describe movement precisely, combine transformations, and predict the resulting position and orientation of objects.