Consider the 3D vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 5 \\ -1 \end{pmatrix}$. Calculate the exact scalar product of these vectors, and hence state the angle between them to 1 decimal place.

1. Calculate the scalar product using the components: $\mathbf{a} \cdot \mathbf{b} = (2)(4) + (-1)(5) + (3)(-1)$. 2. Evaluate the sum: $8 - 5 - 3 = 0$. 3. Identify the formula for the angle between vectors: $\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$. 4. Recognize that since the scalar product is exactly $0$, the entire fraction evaluates to $0$. 5. Conclude that $\cos \theta = 0$, meaning the vectors are perpendicular. The angle is $90.0^\circ$ .

Find the angle between the 3D vectors $u = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $v = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ . Give your answer in degrees correct to 1 decimal place.

$\cos \theta = \frac{u \cdot v}{|u||v|}$ . [M1] $u \cdot v = (1)(3) + (2)(0) + (-1)(4) = 3 + 0 - 4 = -1$ . [A1] $|u| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{6}$ . $|v| = \sqrt{3^2 + 0^2 + 4^2} = \sqrt{25} = 5$ . [A1] $\cos \theta = \frac{-1}{5\sqrt{6}}$ . [M1] $\theta = \cos^{-1}\left(\frac{-1}{5\sqrt{6}}\right) \approx 94.7^\circ$ . [A1]

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IB Maths AI HL · Unit 3: Geometry and Trigonometry

IB Maths AI HL Scalar Products and Voronoi Questions

Updated October 2026

Exam-style IB Maths AI HL scalar products and Voronoi questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

32 questions
Paper 1: 29
Paper 2: 3
10 easy
13 medium
9 hard
3 worked examples

Practise Scalar Products and Voronoi questions →

What you need to know

Voronoi diagrams are SL content (AI SL 3.6) that HL AI students sit too: build cells from perpendicular bisectors, add a new site to an existing diagram, and solve the 'toxic waste dump' problem — the point furthest from every site. Voronoi diagrams — construction and applications overview →

The dot product for angles, cross product for perpendicular vectors. HL AI applies vectors to force decomposition and structural engineering. 3D vectors — dot product and cross product overview →

What's examined in AI HL scalar products and Voronoi

The question bank covers these scalar products and Voronoi question types (number of questions in brackets):

Geometric Matrix Transformations (15)
Scalar Product Calculations (9)
Voronoi Diagram Properties (8)

Key formulas

Vector dot product: $\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$

In the same notation as the IB formula booklet. All AI HL formulas →

Explained, with a worked example: Scalar (dot) product

Where students lose marks on scalar products and Voronoi

Examiner's noteIB notation: AB as a length, vectors in columns, f⁻¹ vs f′, the variables asked for

Where marks are lost: A "perennial problem" in the reports' words: in May 2025 at least a third of candidates read AB (a length) as the vector from A to B. Vectors were written as coordinates, position vectors used where displacement vectors were needed, the inverse-function symbol used for a derivative (and f⁻¹ confused with 1/f), answers given in x and y when the question used other variables, and "h =" left off an equation.

What examiners look for: AB (no arrow) is a length; AB with an arrow is a vector, written as a column. Displacement AB = b − a. f⁻¹ is the inverse, f′ the derivative. Answer in the question's variables.

Examiners flagged this in 3 exam series (May 2024, November 2024, May 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Scalar Products and Voronoi worked examples

Worked example 1: Calculating the Angle Between Vectors · easy

Consider the 3D vectors $a = (\begin{matrix} 2 \\ - 1 \\ 3 \end{matrix})$ and $b = (\begin{matrix} 4 \\ 5 \\ - 1 \end{matrix})$ . Calculate the exact scalar product of these vectors, and hence state the angle between them to 1 decimal place.

Solution

5 steps · 1 to fill in

Examiner tip: If the scalar (dot) product evaluates to exactly zero, the vectors are perpendicular (orthogonal), saving you the valuable exam time of calculating the individual vector magnitudes.

Worked example 2: Finding a Voronoi Boundary Equation · medium

In a Voronoi diagram, two sites are located at $A (2, 5)$ and $B (8, - 1)$ . Find the equation of the boundary separating site $A$ and site $B$ . Give your answer in the form $y = m x + c$ .

Solution

6 steps · 1 to fill in

Examiner tip: Ensure you use the calculated midpoint to form the boundary line equation, not one of the original site coordinates, as the boundary must pass exactly halfway between them.

Worked example 3: Locating a Voronoi Vertex · hard

In a Voronoi diagram for three sites $A$ , $B$ , and $C$ , the boundary between Site $A$ and Site $B$ has the equation $y = 2 x - 4$ . The boundary between Site $B$ and Site $C$ has the equation $x + 3 y = 16$ . Find the exact coordinates of the Voronoi vertex for these three sites.

Solution

6 steps · 1 to fill in

Examiner tip: A Voronoi vertex for three sites represents the centre of a circle that passes through all three sites; it is equidistant from them and is found by intersecting any two of their shared boundary lines.

Try these IB Maths AI HL scalar products and Voronoi questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

The vectors $p = (\begin{matrix} k \\ 3 \end{matrix})$ and $q = (\begin{matrix} 6 \\ - 4 \end{matrix})$ are orthogonal (perpendicular). Find the exact value of the constant $k$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Find the angle between the 3D vectors $u = (\begin{matrix} 1 \\ 2 \\ - 1 \end{matrix})$ and $v = (\begin{matrix} 3 \\ 0 \\ 4 \end{matrix})$ .
Give your answer in degrees correct to 1 decimal place.

Full worked solution Free, no sign-in

$\cos θ = \frac{u \cdot v}{| u | | v |}$ . [M1]
$u \cdot v = (1) (3) + (2) (0) + (- 1) (4) = 3 + 0 - 4 = - 1$ . [A1]
$| u | = \sqrt{1^{2} + 2^{2} + (- 1)^{2}} = \sqrt{6}$ . $| v | = \sqrt{3^{2} + 0^{2} + 4^{2}} = \sqrt{25} = 5$ . [A1]
$\cos θ = \frac{- 1}{5 \sqrt{6}}$ . [M1]
$θ = \cos^{- 1} (\frac{- 1}{5 \sqrt{6}}) \approx {94.7}^{\circ}$ . [A1]

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 1

A transformation matrix is given by $A = (\begin{matrix} 5 & 2 \\ 2 & 2 \end{matrix})$ .
In matrix transformations, invariant lines (lines passing through the origin that map onto themselves) correspond to the eigenvectors of the matrix.
By solving the characteristic equation $det (A - λ I) = 0$ , find the two eigenvalues of matrix $A$ .

Attempt it and see the mark scheme →

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All 32 scalar products and Voronoi questions with mark schemes →

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FAQ

How many IB Maths AI HL scalar products and Voronoi questions are there?

There are 32 exam-style scalar products and Voronoi questions in the AI HL question bank (Paper 1: 29 · Paper 2: 3), graded 10 easy, 13 medium, 9 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is scalar products and Voronoi on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 29 · Paper 2: 3. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI HL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AI HL Scalar Products and Voronoi Questions

Try it first

What you need to know

What's examined in AI HL scalar products and Voronoi

Key formulas

Where students lose marks on scalar products and Voronoi

Scalar Products and Voronoi worked examples

Worked example 1: Calculating the Angle Between Vectors · easy

Solution

Worked example 2: Finding a Voronoi Boundary Equation · medium

Solution

Worked example 3: Locating a Voronoi Vertex · hard

Solution

Try these IB Maths AI HL scalar products and Voronoi questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise scalar products and Voronoi for the exam

FAQ

What else is free here

More AI HL Unit 3 topics