Vectors (AQA A-Level Computer Science): Revision Notes

Vectors

Introduction to vectors

Vectors are mathematical objects that can be represented and used in several different ways, both mathematically and geometrically. In computer science, vectors serve multiple important purposes:

• As a data structure to store collections of values
• As a method for mapping one value to another
• As a way of defining geometric shapes and positions in space

Understanding all three interpretations of vectors is essential for A-Level Computer Science, as they each have practical applications in programming and problem-solving.

Representing vectors as a data structure

When writing programs, vectors can be stored as values within a list structure. This is perhaps the most straightforward way to think about vectors in computing.

Using lists and arrays

Consider the Fibonacci sequence, where each number is the sum of the two preceding ones. The first six values of this sequence can be stored in a vector like this:

fibonacci[0] = 0
fibonacci[1] = 1
fibonacci[2] = 1
fibonacci[3] = 2
fibonacci[4] = 3
fibonacci[5] = 5

This can also be described as a one-dimensional array, where each piece of data is an element in the array that can be accessed by its position (index).

In the table above, the top row shows the index (position) and the bottom row shows the corresponding data value. This structure allows quick access to any element by referencing its index number.

Using dictionaries

A dictionary is a data structure that creates a mapping between a key and a value. We've already seen that we can create sets of real numbers that can be applied to vectors. Using a dictionary structure allows us to call an index, which then acts as a look-up to retrieve the actual values.

The general format of a dictionary is:

{0: Value 1, 1: Value 2, 2: Value 3, 3: Value 4...}

The Fibonacci sequence vector from earlier could be represented in a dictionary as:

{0:0, 1:1, 2:1, 3:2, 4:3, 5:5}

This dictionary structure provides a flexible way to store and retrieve vector data, especially when dealing with non-sequential indices.

Representing vectors as a function

A function is a mathematical construct that maps an input to an output. For example, the function $f(x) = x^2$ takes the value of $x$ and maps it to its square. Vectors can be used to represent this type of function relationship.

When using a vector to represent a function, we define:

• F = the function that creates the vector
• S = the complete set of values that the function can be applied to (the domain)
• R = the potential outputs of the function (the range)

This relationship is written as: $F: S \to R$

The graph shows a visual representation of how a function maps input values to output values. All output values must be drawn from $R$, which is treated as a single field from which the function takes its values.

Representing vectors as arrows

Geometrically, vectors can be visualised as arrows. This representation is particularly useful for understanding vector operations and applications in graphics and physics.

Components of a vector

A vector has two key properties:

• Magnitude: the size or length of the vector
• Direction: the orientation of the vector in space

The direction of the arrow is shown by the arrow head, and the magnitude is represented by the length of the arrow. The starting point of the arrow is called the tail, and the endpoint is called the head.

Plotting vectors on coordinate axes

To quantify both the size and direction of a vector, we plot it on $x$ and $y$ axes. A vector $\mathbf{A}$ can be written in the format $\mathbf{A} = (x, y)$, where $x$ and $y$ are called the components of the vector.

The components represent the distance from the origin $(0, 0)$ on the $x$ and $y$ axes respectively. For example, a vector described as $\mathbf{A} = (4, 3)$ means:

• The $x$-component is $4$ (4 units along the $x$-axis)
• The $y$-component is $3$ (3 units along the $y$-axis)

This is often written in column vector notation as:

$\mathbf{A} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$

This notation helps differentiate vectors from coordinate pairs used to plot points on a graph.

Three-dimensional vectors

Objects in three-dimensional space can be represented using the same method, but with an additional component for the $z$-axis.

In this 3D coordinate system, a vector could be represented as $\mathbf{A} = (x, y, z)$, where:

• $x$ represents the distance along the $x$-axis
• $y$ represents the distance along the $y$-axis
• $z$ represents the distance along the $z$-axis

For example, a 3D vector might be written as $\mathbf{A} = (2, 3, 4)$.

Practical applications

Vectors are extremely useful in computing. For instance, in vector graphics, you can resize an image simply by changing the component values in the vector. This is much more efficient than recalculating every pixel.

The diagram shows how the same star shape can be scaled to different sizes (Scale 2, Scale 3, Scale 4) by changing the vector components. This demonstrates one of the key advantages of vector graphics over bitmap images.

Vector addition

It is possible to add vectors together, which has the effect of translating or displacing the vector. Geometrically, this can be visualised by joining the tail of one vector to the head of another vector.

The diagram shows two vectors $\mathbf{a}$ and $\mathbf{b}$ being added together. Notice that a new point $H$ has been created, which can be used as the head of a new resultant vector $\mathbf{a} + \mathbf{b}$.

Calculating vector addition

The sum of two vectors $\mathbf{A}$ and $\mathbf{B}$ can be calculated as follows:

If:

• $\mathbf{A} = (A_1, A_2, A_3)$
• $\mathbf{B} = (B_1, B_2, B_3)$

Then:

$\mathbf{A} + \mathbf{B} = (A_1 + B_1, A_2 + B_2, A_3 + B_3)$

Scalar-vector multiplication

It is possible to multiply vectors by a number, which has the effect of scaling the vector. The number used is called a scalar, and it represents the amount by which you want to change the scale of the vector.

Understanding scaling

Think of it like zooming in on a map. When you zoom in, you are changing the scale. In the case of a vector, if you scale it by a factor of two, it will have twice the magnitude (length). However, the direction will not change as a result of scaling.

This visual example shows how scaling affects the size of objects while maintaining their shape and orientation.

Scalar multiplication process

The original vector $\mathbf{A} = (3, 2)$. When we multiply this by the scalar $2$, we get:

$\mathbf{B} = 2 \times \mathbf{A} = 2 \times (3, 2) = (6, 4)$

Notice that the tail position and direction do not change - only the magnitude has doubled.

Dot product

The dot product is a process of combining two vectors to calculate a single number (scalar value). It has important applications in computing and mathematics.

Dot product formula

The dot product is calculated using the formula:

$\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y$

The diagram shows vector $\mathbf{A}$ broken down into its components $A_x$ (horizontal) and $A_y$ (vertical).

Calculating the dot product

The process involves:

1. Multiplying the $x$-components together
2. Multiplying the $y$-components together
3. Adding these products

Three-dimensional dot product

This method also works in three dimensions by including the $z$ component in the calculation.

Convex combinations

When two vectors are combined to create a third vector, a special relationship exists between all three vectors. This is called a convex combination.

In the diagram, you can see that a new vector $\mathbf{c}$ has been created at right angles to the other vectors. When these combinations are created, they must follow certain mathematical principles.

Understanding convex hull

A convex combination of vectors is one where the new vector must lie within the vector space of the two vectors from which it is made. The convex hull is a spatial representation of the vector space between two vectors.

This diagram shows multiple vectors emanating from point $A$. Vector $AD$ is created by combining vectors $AC$ and $AB$. Notice the imaginary line between points $B$ and $C$. The new vector must fall within the vector space defined by points $A$, $B$ and $C$ in the diagram. This is a visual representation of the convex hull that represents all the points that make up the vector space.

Notice point $E$, which represents the head for another vector. This falls outside the convex hull and is therefore not a valid convex combination.

Mathematical representation

To perform a convex combination mathematically, you multiply one vector either by a scalar or by another vector. This can be represented as:

$\mathbf{D} = \alpha \mathbf{AB} + \beta \mathbf{AC}$

where:

• $\mathbf{AB}$ and $\mathbf{AC}$ are the two vectors being combined
• $\alpha$ (alpha) and $\beta$ (beta) represent the real numbers that each vector will be multiplied by

Uses of dot product

The dot product has practical applications in computing, particularly for generating parity bits used in error checking.

Generating parity using vectors

Given two vectors $\mathbf{u}$ and $\mathbf{v}$, it is possible to generate parity using the bitwise AND and XOR operations.

Summary tables for GF(2) arithmetic

Arithmetic over GF(2) can be summarised in two small tables:

Multiplication (bitwise AND):

$\times$	0	1
0	0	0
1	0	1

Addition (bitwise XOR):

$+$	0	1
0	0	1
1	1	0