Vectors

This lesson covers vectors as a data structure and mathematical concept, as specified in the OCR A-Level Computer Science (H446) specification, section 1.4. Vectors are used in graphics, physics simulations, machine learning, and many other areas of computing.

---

What is a Vector?

A vector is a mathematical object that has both magnitude (size) and direction. In computing, vectors are represented as ordered lists of numbers.

Vectors vs Scalars

Concept	Description	Example
Scalar	A single number (magnitude only)	Temperature: 20, Speed: 5
Vector	An ordered list of numbers (magnitude and direction)	Position: (3, 4), Velocity: (2, -1, 5)

---

Representing Vectors

Vectors can be represented as lists or arrays of numbers. A vector with n components is called an n-dimensional vector.

Examples

2D vector: v = (3, 4)        -- 2 components
3D vector: w = (1, -2, 5)    -- 3 components
4D vector: u = (1, 0, 3, 7)  -- 4 components

As a Dictionary/Function

A vector can also be thought of as a function that maps indices to values:

• v(0) = 3, v(1) = 4

Python Implementation

# Representing vectors as lists
v = [3, 4]
w = [1, -2, 5]

# Accessing components
print(v[0])    # Output: 3
print(w[2])    # Output: 5

---

Vector Operations

1. Vector Addition

Two vectors of the same dimension can be added by adding their corresponding components.

Formula: If u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), then:

u + v = (u1 + v1, u2 + v2, ..., un + vn)

Worked Example

u = (3, 4, 1) and v = (1, -2, 5)

u + v = (3+1, 4+(-2), 1+5) = (4, 2, 6)

Python

def vector_add(u, v):
    return [u[i] + v[i] for i in range(len(u))]

u = [3, 4, 1]
v = [1, -2, 5]
print(vector_add(u, v))    # Output: [4, 2, 6]

Geometric Interpretation

Adding two 2D vectors is like following one displacement then the other. If you walk 3 steps east and 4 steps north (3, 4), then 1 step east and 2 steps south (1, -2), your total displacement is (4, 2).

---

2. Scalar-Vector Multiplication

Multiplying a vector by a scalar (a single number) scales each component.

Formula: If k is a scalar and v = (v1, v2, ..., vn), then:

kv = (k * v1, k * v2, ..., k * vn)

Worked Example

k = 3, v = (2, -1, 4)

3v = (32, 3(-1), 3*4) = (6, -3, 12)

If k > 1, the vector gets longer. If 0 < k < 1, it gets shorter. If k < 0, the direction reverses.

Python

def scalar_multiply(k, v):
    return [k * v[i] for i in range(len(v))]

v = [2, -1, 4]
print(scalar_multiply(3, v))    # Output: [6, -3, 12]
print(scalar_multiply(0.5, v))  # Output: [1.0, -0.5, 2.0]

---

3. Dot Product (Scalar Product)

The dot product of two vectors produces a scalar (a single number). It measures how aligned two vectors are.

Formula: If u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), then:

u . v = u1v1 + u2v2 + ... + un*vn

Worked Example

u = (3, 4) and v = (2, -1)

u . v = (32) + (4(-1)) = 6 + (-4) = 2

Interpretation

Dot Product Value	Meaning
Positive	Vectors point in roughly the same direction
Zero	Vectors are perpendicular (at right angles)
Negative	Vectors point in roughly opposite directions

Python

def dot_product(u, v):
    return sum(u[i] * v[i] for i in range(len(u)))