Mathematics (Specification B) | Vectors and transformation geometry

Study Notes: Vectors and Transformation Geometry

Welcome to the fascinating world of Vectors and Transformation Geometry! Don't worry if this chapter seems tricky at first; it's all about movement, position, and direction—concepts you use every single day. By understanding vectors and transformations, you'll be able to precisely describe how objects move and change shape. Let's start moving!

1. Understanding Vectors: The GPS of Mathematics

A vector is essentially an instruction that tells you how to move from one point to another. It has two essential ingredients:

• Magnitude: How far to move (the distance).
• Direction: Which way to move.

Analogy: If you tell a friend to walk 3 meters East, you have given them a vector!

1.1 Vector Notation (Column Vectors)

In coordinates, we usually write vectors using column vector notation. This shows the movement in the x-direction (horizontal) and the y-direction (vertical).

A vector \(\mathbf{a}\) moving 3 units right and 4 units up is written as:

\(\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)

• The top number is the movement in the x-direction (positive is right, negative is left).
• The bottom number is the movement in the y-direction (positive is up, negative is down).

Key Term: Displacement Vector
If a point A has coordinates \((x_1, y_1)\) and point B has coordinates \((x_2, y_2)\), the vector \(\vec{AB}\) (the displacement from A to B) is found by subtracting the coordinates:

\(\vec{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}\)

1.2 Vector Operations

Dealing with vectors is just like dealing with coordinates—you work component by component!

Vector Addition and Subtraction

If \(\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\):

Addition: You add the x components and the y components separately.

\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 5 + 3 \end{pmatrix} = \begin{pmatrix} 1 \\ 8 \end{pmatrix}\)

Subtraction: You subtract the components separately.

\(\mathbf{a} - \mathbf{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} - \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 - (-1) \\ 5 - 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\)

Scalar Multiplication

Multiplying a vector by a number (called a scalar) changes its magnitude, but not its direction (unless the scalar is negative, which reverses the direction).

If \(\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\), then \(3\mathbf{a}\) means you multiply both components by 3:

\(3\mathbf{a} = 3 \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times 5 \end{pmatrix} = \begin{pmatrix} 6 \\ 15 \end{pmatrix}\)

Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other. For example, \(\mathbf{p} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) are parallel because \(\mathbf{p} = 2\mathbf{q}\).

1.3 Finding the Magnitude (Length) of a Vector

The magnitude (or length) of a vector is calculated using Pythagoras' Theorem, as the components form a right-angled triangle!

For a vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), the magnitude (written as \(|\mathbf{a}|\)) is:

\(|\mathbf{a}| = \sqrt{x^2 + y^2}\)

Example: Find the magnitude of \(\mathbf{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\)

\(|\mathbf{a}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

2. Transformation Geometry: Changing Position and Shape

A transformation maps points of a shape (the object) onto new points (the image). We focus on four key types:

2.1 Translation (The Simple Shift)

A translation is a simple slide where every point moves the same distance in the same direction. It is described entirely by a vector.

If a point P \((x, y)\) is translated by the vector \(\mathbf{t} = \begin{pmatrix} a \\ b \end{pmatrix}\), the new point P' is:

\(P' = (x + a, y + b)\)

Example: Translate the point (2, 5) by the vector \(\begin{pmatrix} -3 \\ 1 \end{pmatrix}\)

\(New \ Point = (2 + (-3), 5 + 1) = (-1, 6)\)

2.2 Reflection (The Flip)

A reflection flips a shape across a line of reflection (like a mirror).

Key Rules for Standard Mirror Lines:

• Reflection in the x-axis (\(y=0\)): \((x, y) \to (x, -y)\)
• Reflection in the y-axis (\(x=0\)): \((x, y) \to (-x, y)\)
• Reflection in the line \(y=x\): \((x, y) \to (y, x)\) (Swap coordinates)
• Reflection in the line \(y=-x\): \((x, y) \to (-y, -x)\) (Swap and negate)

Common Mistake Alert: Students often mix up reflection in the x-axis (changes y-sign) and reflection in the y-axis (changes x-sign). Think: "Flipping over the x-line changes the y-value's sign."

2.3 Rotation (The Turn)

A rotation turns a shape around a fixed point, called the centre of rotation, through a specific angle and direction (clockwise, CW, or anti-clockwise, ACW).

What you need to describe a rotation:

1. The Centre of rotation (often the origin (0, 0)).
2. The Angle (e.g., 90°, 180°, 270°).
3. The Direction (CW or ACW).

Tip for Rotation: Use tracing paper! Place it over the shape, put your pencil point on the centre of rotation, and turn the paper the required angle/direction. Mark the new position.

2.4 Enlargement (Scaling)

An enlargement changes the size of a shape by multiplying its dimensions by a scale factor (\(k\)) from a fixed centre of enlargement.

What you need to describe an enlargement:

1. The Centre of enlargement (C).
2. The Scale Factor (\(k\)).

Scale Factor Rules:

• If \(k > 1\), the image is larger than the object.
• If \(0 < k < 1\), the image is smaller (a reduction).
• If \(k = 1\), the shape stays the same (Identity).
• If \(k\) is negative (e.g., \(k=-2\)), the shape is enlarged AND flipped through the centre of enlargement (it appears on the opposite side).

Step-by-Step for Drawing Enlargements (Centre Not at Origin):
Let C be the center and P be a point on the object.

1. Find the vector \(\vec{CP}\) (The path from C to P).
2. Multiply this vector by the scale factor \(k\): \(k \times \vec{CP}\).
3. Starting from C, apply the new vector \(k \times \vec{CP}\) to find the image point P'.

Did you know? Enlargement is the only transformation here that changes the area of the shape. If the scale factor is \(k\), the Area Scale Factor is \(k^2\)!

3. Matrix Transformations (The Formal Method)

For Specification B, you must understand how to use \(2 \times 2\) matrices to perform transformations. This is the precise mathematical way to define a movement, especially rotations and reflections centered at the origin.

3.1 The Basics of Matrix Multiplication for Transformations

A matrix transformation takes a point \((x, y)\), written as a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\), and multiplies it by a \(2 \times 2\) transformation matrix \(\mathbf{M}\) to get the new point \((x', y')\).

\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}\)

How to Multiply:

\(x' = (a \times x) + (b \times y)\)
\(y' = (c \times x) + (d \times y)\)

Rule of Thumb: Row times Column.

3.2 Key Transformation Matrices (Centred at the Origin)

These matrices are fundamental and you should recognize or be able to derive them by seeing where the points (1, 0) and (0, 1) land after the transformation.

Transformation	Matrix \(\mathbf{M}\)
Identity (No change)	\(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
Reflection in the x-axis (\(y=0\))	\(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
Reflection in the y-axis (\(x=0\))	\(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
Rotation 90° ACW (Anti-Clockwise)	\(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
Rotation 180° (ACW or CW)	\(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\)
Enlargement, Scale Factor \(k\)	\(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)

Practice Step: Take the matrix for 90° ACW rotation and multiply it by the point (5, 2). You should get (-2, 5). If you can visualize that turn, the matrix works!

3.3 Finding the Area Scale Factor using Matrices

If a transformation is represented by the matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the ratio of the area of the image to the area of the object is given by the determinant of the matrix.

Determinant of \(\mathbf{M}\) (denoted \(|\mathbf{M}|\)) \( = ad - bc\)

Area of Image = Area of Object \(\times\) \(|\mathbf{M}|\)

Note: The absolute value of the determinant is used for area scaling, as area must be positive.

4. Combined Transformations

Sometimes, a shape undergoes a sequence of transformations, like a reflection followed by a rotation. This is called a combined transformation.

4.1 The Importance of Order

The order in which you apply transformations matters! Reflection then rotation gives a different result than rotation then reflection.

If transformation \(T_1\) is followed by transformation \(T_2\), we write this as \(T_2 T_1\). You must perform \(T_1\) first.

4.2 Combined Matrix Multiplication

If transformation \(T_1\) is represented by matrix \(\mathbf{M}_1\) and \(T_2\) by matrix \(\mathbf{M}_2\), the single matrix \(\mathbf{R}\) that represents the combined transformation \(T_2 T_1\) is found by multiplying the matrices in reverse order of operation:

\(\mathbf{R} = \mathbf{M}_2 \mathbf{M}_1\)

Step-by-Step Matrix Combination:

1. Identify the matrix for the first transformation (\(\mathbf{M}_1\)).
2. Identify the matrix for the second transformation (\(\mathbf{M}_2\)).
3. Multiply the second matrix by the first matrix: \(\mathbf{R} = \mathbf{M}_2 \mathbf{M}_1\).
4. Use the resultant single matrix \(\mathbf{R}\) to find the final image of any point.

Memory Aid: When combining matrices, they line up next to the point/vector they are acting on. Since matrix multiplication is done right to left, the matrix closest to the point is the one applied first!

4.3 Inverse Transformations

The inverse transformation, \(T^{-1}\), reverses the effect of the original transformation \(T\). For example, if T is a rotation of 90° ACW, \(T^{-1}\) is a rotation of 90° CW.

For a matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the inverse matrix \(\mathbf{M}^{-1}\) is:

\(\mathbf{M}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

(Where \(ad - bc\) is the determinant).

Don't worry if calculating the inverse looks complicated—just follow the formula steps carefully! The key is finding the determinant and then swapping \(a\) and \(d\), and negating \(b\) and \(c\).