Free A-Level Computer Science guide

Vectors: representations, combinations and dot products

The same vector can be an indexed collection, a function on component indices, or a geometric arrow. Keep those representations consistent while applying one operation at a time.

Content owner: Michael Print · Written for A-Level learners · Checked against official specifications

Explore the idea

Make vector coordinates and geometry agree

A vector has indexed components from one field. Here the field is the real numbers and the dimension is two. Change coordinates and compare addition, scaling and convex combinations.

u[0]u[1]v[0]v[1]Vector operation

u=[3,1], v=[1,3], selected result=[4,4]. A convex result lies on the segment joining u and v; α=0 gives v and α=1 gives u.

Equivalent representations of u ∈ R²
Representation	Data
List / one-dimensional array	[3, 1]
Dictionary indexed by component	{0: 3, 1: 1}
Function on {0,1}	f(0)=3, f(1)=1
Geometric arrow	origin → (3,1)

What is the dot product u · v?

Your answer

Predict before checking or revealing the reasoning.

Find the angle and interpret zero vectors

|u|=3.162, |v|=3.162. Dot product=6; θ=53.13°.

Both vectors must use the same component index set. A scalar multiple changes magnitude and may reverse direction; vector addition translates component by component.

The idea to start with

A vector in R² has two ordered real components, such as [3,1]. A list can store them, a dictionary can map indices 0 and 1 to their values, and a function on {0,1} expresses that same mapping.

Addition and scalar multiplication act component by component. A convex combination uses non-negative coefficients summing to one. The dot product sums matching component products and produces a scalar; with nonzero vectors it can determine their angle.

AQA 7517 · 4.2.8.1

Before you start

Useful foundations

Ordered coordinates and signed arithmetic
Squares, square roots and cosine for angle calculations

By the end, you should be able to

Represent a vector using a list, dictionary, function and geometric arrow.
Add vectors and multiply by scalars over a declared field.
Compute convex combinations and dot products.
Use the dot product to find an angle, excluding zero-vector cases.

Explanations and reference

Board scope and conventions

AQA: vectors over a field, indexed/function/geometric representations, addition, scalar multiplication, convex combinations, dot products and angle application. Examples use real components and exclude zero-vector angle division.

Shared foundations and optional extensions are labelled separately; this page does not claim complete coverage of the qualification.

Keep the field and index set explicit

R² describes two components from the field of real numbers. The vector [3,1] can be represented as {0:3, 1:1}, or f(0)=3 and f(1)=1 on index set {0,1}. Its geometric arrow runs from the origin to (3,1).

A vector need not be two-dimensional; the index set fixes its dimension. Combining vectors componentwise requires compatible dimensions. A geometric point and an arrow from the origin can share coordinates without making every arbitrary record a mathematical vector.

Add and scale componentwise

For u=[3,1] and v=[1,3], u+v=[4,4]. Multiplying u by scalar 2 gives [6,2]. Addition corresponds to translation; positive scaling changes magnitude, negative scaling also reverses direction, and zero scaling gives the zero vector.

Python list addition concatenates rather than performing mathematical vector addition. Build matching component results explicitly, for example [u[i] + v[i] for i in range(len(u))], after checking dimensions.

A convex combination stays between two endpoints

For 0 ≤ α ≤ 1, αu+(1−α)v has coefficients that are non-negative and sum to one. At α=0 it is v; at α=1 it is u. Intermediate weights lie on the segment connecting their coordinate points.

With α=0.25, u=[3,1] and v=[1,3], the result is [1.5,2.5]. Coefficients 2 and −1 still sum to one but are not a convex combination because one coefficient is negative.

Use the dot product for an angle

u·v = Σ uᵢvᵢ. Here 3×1 + 1×3 = 6. Magnitudes are |u|=sqrt(10) and |v|=sqrt(10), so cos θ=6/10=0.6 and θ≈53.13°. Use the angle unit required by the question.

If either vector is zero, dividing by |u||v| is undefined. A dot product of zero establishes perpendicularity only when both vectors are nonzero. Rounding intermediate magnitudes too early can also distort an angle.

One pair, four different results

Worked example

One pair, four different results

Let u=[2,0] and v=[0,2]. u+v=[2,2]; 3u=[6,0].
With weights 0.5 and 0.5, the convex combination is [1,1]. It lies halfway along the segment between the two endpoint coordinates.
The dot product is 2×0 + 0×2 = 0. Both vectors have magnitude 2, so cos θ=0 and θ=90°.
Replacing u by [0,0] still gives a dot product of zero, but the angle is now undefined; do not label the zero arrow perpendicular.

Common mix-up: A dot product is another vector

Common mix-up: Weights summing to one are always convex

Original A-Level practice

4 original questions total 12 marks. Attempt each before opening the independently written indicative marking guidance.

Question 1

3 marks

Let u=[2,−1] and v=[4,3]. Give a dictionary representation of u, calculate u+v and calculate −2u.

Show solution and marking guidance

Indicative answer

{0:2, 1:-1}.
u+v=[6,2].
−2u=[−4,2].

Question 2

3 marks

For u=[2,0], v=[0,4], compute 0.25u+0.75v. Explain whether 2u−v is a convex combination.

Show solution and marking guidance

Indicative answer

[0.5,3].
The first coefficients are non-negative and sum to one.
2u−v is not convex because the coefficient of v is negative.

Question 3

3 marks

For u=[1,1] and v=[1,−1], find the dot product and the angle. State the nonzero condition required by the angle formula.

Show solution and marking guidance

Indicative answer

Dot product = 1−1 = 0.
Both magnitudes are sqrt(2), so the angle is 90°.
Neither vector may have zero magnitude.

Question 4

3 marks

Explain why Python [2,1] + [3,4] does not perform vector addition. Give the mathematical result and the Python list result.

Show solution and marking guidance

Indicative answer

Python + concatenates lists.
Mathematical addition gives [5,5].
The Python expression gives [2,1,3,4].

Specification and references

This guide addresses AQA 7517 4.2.8.1. Check your examination year and the complete specification for the assessment scope.

AQA 7517 official subject content; checked 23 October 2026

These are independently written explanations and practice questions. CompSciTutoring.co.uk is not affiliated with or endorsed by an examination board. The marking guidance is indicative; always check the syllabus for your examination year.

Found an error? Email a correction.

Need help applying this to unfamiliar questions?

Use the free examples first, then explore one-to-one tuition if guided practice would help.

Explore tuition Email Michael

Free A-Level Computer Science guide

Vectors: representations, combinations and dot products

The same vector can be an indexed collection, a function on component indices, or a geometric arrow. Keep those representations consistent while applying one operation at a time.

Content owner: Michael Print · Written for A-Level learners · Checked against official specifications

Explore the idea

Make vector coordinates and geometry agree

A vector has indexed components from one field. Here the field is the real numbers and the dimension is two. Change coordinates and compare addition, scaling and convex combinations.

u[0]u[1]v[0]v[1]Vector operation

u=[3,1], v=[1,3], selected result=[4,4]. A convex result lies on the segment joining u and v; α=0 gives v and α=1 gives u.

Equivalent representations of u ∈ R²
Representation	Data
List / one-dimensional array	[3, 1]
Dictionary indexed by component	{0: 3, 1: 1}
Function on {0,1}	f(0)=3, f(1)=1
Geometric arrow	origin → (3,1)

What is the dot product u · v?

Your answer

Predict before checking or revealing the reasoning.

Find the angle and interpret zero vectors

|u|=3.162, |v|=3.162. Dot product=6; θ=53.13°.

Both vectors must use the same component index set. A scalar multiple changes magnitude and may reverse direction; vector addition translates component by component.

The idea to start with

A vector in R² has two ordered real components, such as [3,1]. A list can store them, a dictionary can map indices 0 and 1 to their values, and a function on {0,1} expresses that same mapping.

AQA 7517 · 4.2.8.1

Before you start

Useful foundations

Ordered coordinates and signed arithmetic
Squares, square roots and cosine for angle calculations

By the end, you should be able to

Represent a vector using a list, dictionary, function and geometric arrow.
Add vectors and multiply by scalars over a declared field.
Compute convex combinations and dot products.
Use the dot product to find an angle, excluding zero-vector cases.

Explanations and reference

Board scope and conventions

Shared foundations and optional extensions are labelled separately; this page does not claim complete coverage of the qualification.

Keep the field and index set explicit

Add and scale componentwise

A convex combination stays between two endpoints

For 0 ≤ α ≤ 1, αu+(1−α)v has coefficients that are non-negative and sum to one. At α=0 it is v; at α=1 it is u. Intermediate weights lie on the segment connecting their coordinate points.

With α=0.25, u=[3,1] and v=[1,3], the result is [1.5,2.5]. Coefficients 2 and −1 still sum to one but are not a convex combination because one coefficient is negative.

Use the dot product for an angle

u·v = Σ uᵢvᵢ. Here 3×1 + 1×3 = 6. Magnitudes are |u|=sqrt(10) and |v|=sqrt(10), so cos θ=6/10=0.6 and θ≈53.13°. Use the angle unit required by the question.

One pair, four different results

Worked example

One pair, four different results

Let u=[2,0] and v=[0,2]. u+v=[2,2]; 3u=[6,0].
With weights 0.5 and 0.5, the convex combination is [1,1]. It lies halfway along the segment between the two endpoint coordinates.
The dot product is 2×0 + 0×2 = 0. Both vectors have magnitude 2, so cos θ=0 and θ=90°.
Replacing u by [0,0] still gives a dot product of zero, but the angle is now undefined; do not label the zero arrow perpendicular.

Common mix-up: A dot product is another vector

Common mix-up: Weights summing to one are always convex

Original A-Level practice

4 original questions total 12 marks. Attempt each before opening the independently written indicative marking guidance.

Question 1

3 marks

Let u=[2,−1] and v=[4,3]. Give a dictionary representation of u, calculate u+v and calculate −2u.

Show solution and marking guidance

Indicative answer

{0:2, 1:-1}.
u+v=[6,2].
−2u=[−4,2].

Question 2

3 marks

For u=[2,0], v=[0,4], compute 0.25u+0.75v. Explain whether 2u−v is a convex combination.

Show solution and marking guidance

Indicative answer

[0.5,3].
The first coefficients are non-negative and sum to one.
2u−v is not convex because the coefficient of v is negative.

Question 3

3 marks

For u=[1,1] and v=[1,−1], find the dot product and the angle. State the nonzero condition required by the angle formula.

Show solution and marking guidance

Indicative answer

Dot product = 1−1 = 0.
Both magnitudes are sqrt(2), so the angle is 90°.
Neither vector may have zero magnitude.

Question 4

3 marks

Explain why Python [2,1] + [3,4] does not perform vector addition. Give the mathematical result and the Python list result.

Show solution and marking guidance

Indicative answer

Python + concatenates lists.
Mathematical addition gives [5,5].
The Python expression gives [2,1,3,4].

Specification and references

This guide addresses AQA 7517 4.2.8.1. Check your examination year and the complete specification for the assessment scope.

AQA 7517 official subject content; checked 23 October 2026

Found an error? Email a correction.

Need help applying this to unfamiliar questions?

Use the free examples first, then explore one-to-one tuition if guided practice would help.

Explore tuition Email Michael