IB Maths AA HL · Unit 3: Geometry and Trigonometry
IB Maths AA HL Planes Questions
Exam-style IB Maths AA HL planes 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.
- 31 questions
- Paper 1: 23
- Paper 2: 8
- 7 easy
- 13 medium
- 7 hard
- 4 starter
- 3 worked examples
Practise Planes questions →
AA HL formula booklet
What's examined in AA HL planes
The question bank covers these planes question types (number of questions in brackets):
- 3D Geometry Problems (18)
- Plane Equations (7)
- Vector Definitions & Operations (6)
Key formulas
- Vector magnitude
- \(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\)
- Vector equation of a line
- \(\vec{r} = \vec{a} + t\,\vec{d}\)
- Cartesian equation of a plane
- \(\vec{r}\cdot\vec{n} = \vec{a}\cdot\vec{n}\)
In the same notation as the IB formula booklet. All AA HL formulas →
Planes worked examples
Worked example 1: Finding the Cartesian equation of a plane · easy
A plane $\Pi$ passes through the point $A(1, 2, -1)$ and has a normal vector $\mathbf{n} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$. Find the Cartesian equation of the plane.
1. Identify the standard scalar product form for the equation of a plane: $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$.
2. Substitute the position vector of point $A$ and the normal vector into the right-hand side: $\mathbf{a} \cdot \mathbf{n} = (1)(3) + (2)(-1) + (-1)(2)$.
3. Calculate the scalar product constant: $3 - 2 - 2 = -1$.
4. Expand the left-hand side $\mathbf{r} \cdot \mathbf{n}$ using $\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ to get $3x - y + 2z$.
5. State the final Cartesian equation: $\mathbf{3x - y + 2z = -1}$.
Examiner tip: Don't forget that the coefficients of $x$, $y$, and $z$ in a Cartesian plane equation are exactly the components of the normal vector; you can use this to quickly reverse-engineer a normal vector if given the plane equation.
Worked example 2: Intersection of a line and a plane · medium
Find the exact coordinates of the point of intersection between the line $L: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$ and the plane $\Pi: 2x + y - z = 5$.
1. Express the line equations parametrically in terms of $t$: $x = t$, $y = 1 - t$, and $z = 2 + 3t$.
2. Substitute these parametric expressions for $x$, $y$, and $z$ directly into the Cartesian equation of the plane: $2(t) + (1 - t) - (2 + 3t) = 5$.
3. Expand the brackets carefully: $2t + 1 - t - 2 - 3t = 5$.
4. Simplify and collect like terms to solve for the parameter $t$: $-2t - 1 = 5 \implies -2t = 6 \implies t = -3$.
5. Substitute $t = -3$ back into the original parametric equations of the line: $x = -3$, $y = 1 - (-3) = 4$, $z = 2 + 3(-3) = -7$.
6. State the final exact coordinates of the intersection point: $\mathbf{(-3, 4, -7)}$.
Examiner tip: A frequent error is finding the value of the parameter $t$ and stopping there; always remember to substitute $t$ back into the line equation to find the actual 3D coordinates requested by the question.
Worked example 3: Finding the acute angle between planes · hard
Find the acute angle between the planes $\Pi_1: x + 2y - z = 4$ and $\Pi_2: 2x - y + 3z = 7$. Give your answer in degrees to one decimal place.
1. Extract the normal vectors for each plane from their Cartesian coefficients: $\mathbf{n}_1 = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{n}_2 = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$.
2. Calculate the scalar product of the normal vectors: $\mathbf{n}_1 \cdot \mathbf{n}_2 = (1)(2) + (2)(-1) + (-1)(3) = 2 - 2 - 3 = -3$.
3. Find the magnitude of each normal vector: $|\mathbf{n}_1| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{6}$ and $|\mathbf{n}_2| = \sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{14}$.
4. Substitute the values into the rearranged scalar product angle formula, taking the absolute value of the dot product to guarantee an acute angle: $\cos \theta = \frac{|-3|}{\sqrt{6}\sqrt{14}} = \frac{3}{\sqrt{84}}$.
5. Evaluate the inverse cosine using your GDC: $\theta = \arccos\left(\frac{3}{\sqrt{84}}\right) = 70.893...^\circ$.
6. State the final angle rounded to one decimal place: $\mathbf{70.9^\circ}$.
Examiner tip: Always use the absolute value of the dot product ($|\mathbf{n}_1 \cdot \mathbf{n}_2|$) when finding the acute angle between planes; omitting this step when the dot product is negative will result in you calculating the obtuse angle instead.
Try these IB Maths AA HL planes 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 · 2 marks · Paper 1
Find the scalar (dot) product \(\mathbf{a} \cdot \mathbf{b}\) where \(\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}\).
Question 2 · medium · 5 marks · Paper 2
Particle A starts at \((0, 2, 1)\) and travels with velocity \(\mathbf{v}_A = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\) m/s.
Particle B starts at \((3, 0, 0)\) and travels with velocity \(\mathbf{v}_B = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\) m/s.
Find the exact distance between the two particles after \(t = 3\) seconds.
Question 3 · hard · 5 marks · Paper 2
Three planes are defined by the following Cartesian equations:
\[\Pi_1: x + y + z = 6\]
\[\Pi_2: 2x - y + z = 3\]
\[\Pi_3: x + 2y - z = 2\]
Show that the three planes intersect at a single unique point, and find the coordinates of this point.
All 31 planes questions with mark schemes →
FAQ
How many IB Maths AA HL planes questions are there?
There are 31 exam-style planes questions in the AA HL question bank (Paper 1: 23 · Paper 2: 8), graded 7 easy, 13 medium, 7 hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is planes on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 23 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA 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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