IB Maths AI HL · Unit 3: Geometry and Trigonometry
IB Maths AI HL Scalar Products and Voronoi Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 19 questions
- Paper 1: 19
- 5 easy
- 5 medium
- 4 hard
- 5 starter
- 3 worked examples
Practise Scalar Products and Voronoi questions →
AI HL formula booklet
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):
- Scalar Product Calculations (7)
- Geometric Matrix Transformations (7)
- Voronoi Diagram Properties (5)
Key formulas
- Vector dot product
- \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3\)
In the same notation as the IB formula booklet. All AI HL formulas →
Scalar Products and Voronoi worked examples
Worked example 1: Calculating the Angle Between Vectors · easy
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 $90.0^\circ$.
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 $y = mx + c$.
1. Recall that a Voronoi boundary is the perpendicular bisector of the line segment connecting the two sites.
2. Find the midpoint of $AB$: $M = \left( \frac{2+8}{2}, \frac{5+(-1)}{2} \right) = (5, 2)$.
3. Calculate the gradient of $AB$: $m_{AB} = \frac{-1-5}{8-2} = \frac{-6}{6} = -1$.
4. Determine the perpendicular gradient for the boundary: $m_{\perp} = \frac{-1}{-1} = 1$.
5. Substitute the midpoint and the perpendicular gradient into the point-slope formula: $y - 2 = 1(x - 5)$.
6. Rearrange into the required form: $y = x - 5 + 2$, so $y = x - 3$.
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 $y = 2x - 4$. The boundary between Site $B$ and Site $C$ has the equation $x + 3y = 16$. Find the exact coordinates of the Voronoi vertex for these three sites.
1. Recognize that a Voronoi vertex is the point of intersection of the boundaries shared by the adjacent sites.
2. Set up the system of linear equations: $y = 2x - 4$ and $x + 3y = 16$.
3. Substitute the first equation directly into the second to eliminate $y$: $x + 3(2x - 4) = 16$.
4. Expand and solve for $x$: $x + 6x - 12 = 16 \implies 7x = 28 \implies x = 4$.
5. Substitute $x = 4$ back into the first equation to find $y$: $y = 2(4) - 4 = 4$.
6. The exact coordinates of the Voronoi vertex are $(4, 4)$.
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.
Question 1 · easy · 4 marks · Paper 1
The vectors \(p = \begin{pmatrix} k \\ 3 \end{pmatrix}\) and \(q = \begin{pmatrix} 6 \\ -4 \end{pmatrix}\) are orthogonal (perpendicular). Find the exact value of the constant \(k\).
Question 2 · medium · 5 marks · Paper 1
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.
Question 3 · hard · 7 marks · Paper 1
A transformation matrix is given by \(A = \begin{pmatrix} 5 & 2 \\ 2 & 2 \end{pmatrix}\).
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 \(\det(A - \lambda I) = 0\), find the two eigenvalues of matrix \(A\).
All 19 scalar products and voronoi questions with mark schemes →
FAQ
How many IB Maths AI HL scalar products and voronoi questions are there?
There are 19 exam-style scalar products and voronoi questions in the AI HL question bank (Paper 1: 19), graded 5 easy, 5 medium, 4 hard, 5 starter. 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?
In the question bank these questions are set as Paper 1 questions.
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.
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