IB Maths AI SL · Unit 3: Geometry and Trigonometry
Exam-style IB Maths AI SL 3d coordinate geometry 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.
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Find the exact distance between $A(1, 4, -2)$ and $B(3, 5, 2)$.
1. Recall: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
2. Substitute: $d = \sqrt{(3-1)^2 + (5-4)^2 + (2-(-2))^2}$.
3. Simplify: $d = \sqrt{2^2 + 1^2 + 4^2}$.
4. Evaluate: $d = \sqrt{4 + 1 + 16} = \mathbf{\sqrt{21}}$ units.
A drone flies from $P(-2, 10, 5)$ to $Q(6, -4, 13)$. Find the exact midpoint.
1. Recall: $M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2},\ \frac{z_1+z_2}{2}\right)$.
2. x: $\frac{-2+6}{2} = 2$.
3. y: $\frac{10+(-4)}{2} = 3$.
4. z: $\frac{5+13}{2} = 9$.
5. State: $\mathbf{(2, 3, 9)}$.
The distance between $C(0, 1, -1)$ and $D(2, 3, z)$ is exactly $3$ units. Find both values of $z$.
1. Set up: $\sqrt{4+4+(z+1)^2} = 3$.
2. Square: $8 + (z+1)^2 = 9$.
3. Isolate: $(z+1)^2 = 1$.
4. Both roots: $z+1 = \pm 1$.
5. Solve: $\mathbf{z = 0 \text{ or } z = -2}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
The points \(P(-1, 3, 5)\) and \(Q(5, 7, -1)\) represent the endpoints of a tunnel in a 3D grid.
A rectangular room has dimensions: length \(8 \text{ m}\), width \(5 \text{ m}\), and height \(4 \text{ m}\).
A rectangular cuboid sits on a 3D coordinate system. Its vertices include the origin \(O(0,0,0)\), and the points \(A(5,0,0)\), \(B(0,4,0)\), and \(C(0,0,3)\). Let point \(D\) be the vertex diagonally opposite to the origin.
There are 12 exam-style 3d coordinate geometry questions in the AI SL question bank (Paper 1: 10 · Paper 2: 2), graded 2 very hard, 2 hard, 5 medium, 3 easy. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 10 · Paper 2: 2. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL 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.