Vector Transformations 1

Vector Transformations and Matrix Animation

IB Mathematics HL • Vectors and Matrices • Transformations • Iteration and Invariant Points

Lesson Overview

This lesson focuses on matrix transformations in the plane. In IB Paper 2, students are often asked to interpret a matrix geometrically, combine transformations, apply repeated transformations, and identify invariant or limiting points.

In this question, Amira is creating animations for a website. She uses matrices to transform objects relative to the origin and studies what happens when these transformations are repeated many times.

⭐ Key Concepts

A matrix can represent a rotation, reflection, enlargement, or shear.
Repeated multiplication by the same matrix corresponds to repeated transformations.
If a matrix represents a rotation, powers of the matrix repeat in cycles.
An enlargement with scale factor $k$ about the origin is represented by

$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$

An invariant point $Q$ satisfies

$TQ=Q$

To find a limiting point under repeated transformation, solve the invariant equation.

📘 Clear IB-Style Explanation

When a matrix represents a rotation, its powers are often easiest to understand by thinking geometrically instead of multiplying repeatedly.

If the rotation is $90^\circ$ , then after four applications the object returns to its original position. This helps when finding the smallest value of $n$ such that $A^n=I$ .

In iterative transformation questions, a point can move along a spiral path if each step combines a rotation with an enlargement or contraction.

For the final part, if all points move towards one fixed point after many transformations, that point must satisfy

$TQ=Q$

so it can be found by solving a system of simultaneous equations.

📌 Worked Example — Matrix Transformations

Amira is creating animations for a website. She uses matrices to transform objects relative to the origin, $O$ .

One matrix that she uses is $A$ .

(a) Describe fully the transformation represented by matrix $A$ .

(b) Find the smallest value of $n$ such that $A^n=I$ .

Amira also uses matrix $B$ , which represents an enlargement with scale factor $2$ , centre $O$ .

(c.i) Write down matrix $B$ .

(c.ii) Describe fully the transformation represented by $B^n$ , where $n$ is the value found in part (b).

Amira creates a new matrix, $C=AB$ .

(d) Find matrix $C$ .

Amira creates an animation by repeatedly transforming an object by $C$ .

A point, $P$ , on the object is initially at $(1,0)$ . Amira sets the speed of the animation to 6 transformations per second.

(e) Sketch the path of $P$ for the first 4 seconds of motion and label the coordinates of the start and end points.

Amira uses a different transformation, $T$ , defined by

$T(\mathbf{x})=M\mathbf{x}+\mathbf{v}$

To create her animation, she repeatedly transforms an object by $T$ . After many transformations, she notices that all points, $P$ , on the object tend towards a single point, $Q$ , such that

$TQ=Q$

(f) Find $Q$ , where $Q=(x,y)$ .

Solution

(a) Transformation represented by $A$

Matrix $A$ represents an

$\text{anticlockwise rotation of }90^\circ\text{ about the origin}$

📊 Marks: A1

(b) Smallest value of $n$ such that $A^n=I$

A $90^\circ$ rotation repeated four times gives one full turn:

$4\times 90^\circ=360^\circ$

Therefore the smallest value is

$n=4$

📊 Marks: M1 A1

(c.i) Matrix $B$

An enlargement with scale factor $2$ , centre $O$ , is represented by

$B= \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$

📊 Marks: A1

(c.ii) Transformation represented by $B^n$

Since $n=4$ ,

$B^4= \begin{pmatrix} 16 & 0 \\ 0 & 16 \end{pmatrix}$

So $B^4$ represents an

$\text{enlargement with scale factor }16,\text{ centre }O$

📊 Marks: A1 A1

(d) Matrix $C$

Since $C=AB$ , multiplying gives

$C= \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$

📊 Marks: A2

(e) Path of $P$

Each transformation applies a $90^\circ$ anticlockwise rotation together with an enlargement of scale factor $2$ . Therefore the point moves in an

$\text{anticlockwise spiral}$

The point starts at

$(1,0)$

After $4$ seconds, the number of transformations is

$6\times4=24$

Every 4 transformations the direction repeats, and the distance from the origin is multiplied by $2^4=16$ . After 24 transformations, there are 6 full cycles, so the direction is the same as the start and the scale factor is

$2^{24}$

Therefore the end point is

$\left(2^{24},0\right)$

📊 Marks: A1 A1

(f) Find $Q$ where $TQ=Q$

The limiting point satisfies the invariant equation

$TQ=Q$

Solving the simultaneous equations from the given transformation gives

$Q= \begin{pmatrix} 4 \\ 2 \end{pmatrix}$

Hence

$Q=(4,2)$

📊 Marks: M1 A1 A1

⚠ Common Mistakes

Describing the rotation correctly but forgetting to include the centre.
Using $n=2$ instead of $n=4$ for a $90^\circ$ rotation.
Raising the scale factor incorrectly when interpreting $B^n$ .
Sketching a circle instead of a spiral in part (e).
Trying to estimate the invariant point instead of solving $TQ=Q$ algebraically.

📘 IB Exam Tips

Interpret matrix powers geometrically whenever possible.
For repeated enlargements, remember to raise the scale factor to the required power.
In sketch questions, label both the starting point and the final point clearly.
For invariant points, write down the equation $TQ=Q$ before solving.
Keep matrix multiplication neat, because one sign error can change the whole transformation.

Vector Transformations and Matrix Animation

Lesson Overview

⭐ Key Concepts

📘 Clear IB-Style Explanation

📌 Worked Example — Matrix Transformations

Solution

⚠ Common Mistakes

📘 IB Exam Tips

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Vector Transformations 1

Vector Transformations and Matrix Animation

Lesson Overview

⭐ Key Concepts

📘 Clear IB-Style Explanation

📌 Worked Example — Matrix Transformations

Solution

⚠ Common Mistakes

📘 IB Exam Tips

Related

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