1.4.2f Vectors — OCR H446

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Vectors in Computer Science

In OCR H446, a vector is a mathematical object representing a quantity with both magnitude and direction, represented as an ordered list of numbers (components). Vectors are used extensively in computer graphics, physics simulations, machine learning, and data science.

A vector can also refer to a 1D array of numbers in computing. Both definitions are relevant — the mathematical properties directly inform how they are implemented and used.

Vector Notation and Representation

An n-dimensional vector is written as an ordered tuple of n real numbers:

-- 2D vector (x, y components):
v = (3, 4)   -- 3 units right, 4 units up

-- 3D vector (x, y, z components):
u = (1, 2, 5)

-- n-dimensional vector:
v = (v₁, v₂, v₃, ..., vₙ)

Vectors can also be written as column vectors (matrices with one column) or as lists/arrays in code.

Vector Operations

1. Vector Addition

Add corresponding components: (a₁, a₂) + (b₁, b₂) = (a₁+b₁, a₂+b₂)

(3, 4) + (1, -2) = (3+1, 4+(-2)) = (4, 2)

2. Scalar Multiplication

Multiply each component by a scalar k: k × (a₁, a₂) = (k×a₁, k×a₂)

3 × (2, 5) = (6, 15)
-1 × (3, 4) = (-3, -4)   -- reverses direction

3. Dot Product (Scalar Product)

Produces a scalar: a · b = a₁×b₁ + a₂×b₂ + ... + aₙ×bₙ

-- Example:
(3, 4) · (1, 2) = (3×1) + (4×2) = 3 + 8 = 11

-- Geometric interpretation:
a · b = |a| × |b| × cos(θ)
-- where θ is the angle between vectors
-- If a · b = 0, vectors are perpendicular (cos 90° = 0)

4. Vector Magnitude (Length)

The magnitude (or length/norm) of vector v = (a, b) is:

|v| = √(a² + b²)           -- 2D
|v| = √(a² + b² + c²)     -- 3D

-- Example: |(3, 4)| = √(9 + 16) = √25 = 5

5. Unit Vector (Normalisation)

A unit vector has magnitude 1. To normalise vector v: divide each component by |v|.

v = (3, 4),  |v| = 5
unit_v = (3/5, 4/5) = (0.6, 0.8)
-- Check: √(0.6² + 0.8²) = √(0.36+0.64) = √1 = 1 ✓

Convex Combination of Vectors

A convex combination of two vectors u and v is: w = (1−t)u + tv, where 0 ≤ t ≤ 1. As t goes from 0 to 1, w moves linearly from u to v — this is used for interpolation in graphics (e.g. blending colours, smooth animation transitions).

-- Midpoint (t = 0.5):
u = (2, 4), v = (8, 0)
w = 0.5 × (2,4) + 0.5 × (8,0) = (1,2) + (4,0) = (5, 2)

Vectors in Computing Applications

Application	How Vectors are Used
Computer graphics (2D/3D)	Position vectors, velocity, direction of light, surface normals
Physics engines	Force vectors, acceleration, velocity; vector addition for resultant force
Machine learning	Feature vectors (each dimension = one feature); word embeddings
Computer vision	Image as vector of pixel values; distance between image vectors for similarity
GPS/Navigation	Position and displacement vectors; finding shortest path direction
Cryptography	Lattice-based cryptography uses high-dimensional vectors

Representing Vectors in Code

Vectors are typically implemented as one-dimensional arrays or lists:

-- Python example:
v = [3, 4]
u = [1, 2]

-- Addition:
result = [v[i] + u[i] for i in range(len(v))]   -- [4, 6]

-- Dot product:
dot = sum(v[i] * u[i] for i in range(len(v)))    -- 11

-- Magnitude:
import math
mag = math.sqrt(sum(x**2 for x in v))            -- 5.0

Summary of Operations

Operation	Formula	Result type
Addition	(a₁+b₁, a₂+b₂, ...)	Vector
Scalar multiplication	(k×a₁, k×a₂, ...)	Vector
Dot product	a₁b₁ + a₂b₂ + ... + aₙbₙ	Scalar
Magnitude	√(a₁² + a₂² + ... + aₙ²)	Scalar
Normalisation	v / \|v\|	Vector (unit vector)
Convex combination	(1-t)u + tv, 0 ≤ t ≤ 1	Vector (on line between u,v)

Exam tip: The dot product of two vectors is a SCALAR (single number), not another vector. The formula is: sum of products of corresponding components. The dot product is zero when vectors are perpendicular (angle = 90°). This property is used in 3D graphics for lighting calculations (how much a surface faces a light source).

Exam tip: A unit vector has magnitude exactly 1. To find the unit vector, divide each component by the original vector's magnitude. The magnitude of a 2D vector (a, b) = √(a² + b²) — Pythagoras' theorem in n dimensions.

⚠ Common Mistakes

The dot product gives a scalar, not a vector. Cross product (not in H446) gives a vector. In exams: "dot product of (2,3) and (4,1) = 2×4 + 3×1 = 11" — a number.
Forgetting the square root when computing magnitude — |(3,4)| = 5, not 25.
Confusing vectors (magnitude + direction) with scalars (magnitude only). Speed is scalar; velocity is vector.

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Worksheet — 1.4.2f Vectors

8 questions · 20 marks · instantly marked

Q1Define a vector in computing. State two properties that distinguish a vector from a scalar.[3 marks]

✓ Mark scheme

A vector is a mathematical object representing a quantity with both magnitude (size) and direction, stored as an ordered list of n numeric components [1]. Properties distinguishing it from a scalar: (1) vectors have direction, scalars do not [1]; (2) vectors are ordered tuples of multiple components; scalars are single numbers [1]. Example: speed is scalar, velocity is vector.

Q2Calculate: (a) (5, 2) + (−1, 7) and (b) 4 × (3, −2).[2 marks]

✓ Mark scheme

(a) (5+(−1), 2+7) = (4, 9) [1]. (b) (4×3, 4×(−2)) = (12, −8) [1].

Q3Calculate the dot product of (2, 5) and (4, −1). What type of value does the dot product always produce?[3 marks]

✓ Mark scheme

(2×4) + (5×(−1)) = 8 + (−5) = 3 [2]. The dot product always produces a scalar (a single number, not a vector) [1].

Q4Calculate the magnitude of vector (6, 8). Show your working.[2 marks]

✓ Mark scheme

|v| = √(6² + 8²) = √(36 + 64) = √100 = 10 [2]. Method mark for √(a²+b²) [1], answer mark for 10 [1].

Q5Find the unit vector in the direction of (0, 5). Verify your answer has magnitude 1.[3 marks]

✓ Mark scheme

|v| = √(0² + 5²) = √25 = 5 [1]. Unit vector = (0/5, 5/5) = (0, 1) [1]. Verification: √(0² + 1²) = √1 = 1 ✓ [1].

Q6Calculate the convex combination of u = (4, 0) and v = (0, 8) at t = 0.25. What does this operation represent geometrically?[3 marks]

✓ Mark scheme

w = (1−0.25)(4,0) + 0.25(0,8) = 0.75(4,0) + 0.25(0,8) = (3,0) + (0,2) = (3, 2) [2]. Geometrically: the convex combination finds a point on the straight line between u and v. At t=0.25, the point is 25% of the way from u to v [1]. Used in graphics for interpolation/animation.

Q7Two vectors a = (1, 0) and b = (0, 1) are perpendicular. Verify this using the dot product.[2 marks]

✓ Mark scheme

a · b = (1×0) + (0×1) = 0 + 0 = 0 [1]. When the dot product is 0, the vectors are perpendicular (cos 90° = 0, so a · b = |a||b|cos(90°) = 0) [1].

Q8Give two real-world computing applications where vectors are used, explaining briefly how they are applied in each.[4 marks]

✓ Mark scheme

Any two of [2 marks each]: Computer graphics — 3D position and direction of objects represented as vectors; lighting calculations use dot product to determine how much a surface faces a light source [2]. Machine learning — feature vectors (each dimension = one feature of a data item); similarity between items measured by dot product or distance between vectors [2]. Physics engines — velocity, force, and acceleration represented as vectors; vector addition used to calculate resultant force on an object [2]. GPS/navigation — displacement vectors for calculating direction and distance between coordinates [2].

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Term	Definition

Vectors in Computer Science

Vector Notation and Representation

Vector Operations

1. Vector Addition

2. Scalar Multiplication

3. Dot Product (Scalar Product)

4. Vector Magnitude (Length)

5. Unit Vector (Normalisation)

Convex Combination of Vectors

Vectors in Computing Applications

Representing Vectors in Code

Summary of Operations

Worksheet — 1.4.2f Vectors

Mini Test — 1.4.2f Vectors