3.15 Coincident, parallel, intersecting and skew lines questions for IB Maths AA

22 questions on Coincident, parallel, intersecting and skew lines, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 3 Geometry and trigonometry
HL only

Questions by paper and level

Paper	Questions
Paper 1	10
Paper 2	11
Paper 3	1
Every paper	22

Counted on 10 November 2026

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Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00253Paper 1 · 5 marks

The lines $L_{1}$ and $L_{2}$ have the following vector equations.

L_{1} : r = (\begin{array}{c} 1 \\ - 2 \\ 3 \end{array}) + λ (\begin{array}{c} 2 \\ 1 \\ - 1 \end{array}), λ \in R

L_{2} : r = (\begin{array}{c} 3 \\ - 6 \\ - 3 \end{array}) + μ (\begin{array}{c} 1 \\ 3 \\ 2 \end{array}), μ \in R

Show that $L_{1}$ and $L_{2}$ intersect, and find the coordinates of their point of intersection.

5 marks

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Question 2 · AA-00244Paper 2 · 6 marks

Two drones, A and B, are flying in straight lines with constant velocities. Relative to a fixed origin, their position vectors $t$ seconds after they are launched are given by

r_{A} = (\begin{array}{c} 2 \\ 1 \\ 0 \end{array}) + t (\begin{array}{c} 3 \\ 2 \\ 1 \end{array}) and r_{B} = (\begin{array}{c} 11 \\ - 5 \\ 12 \end{array}) + t (\begin{array}{c} 0 \\ 4 \\ - 3 \end{array}),

where distances are measured in metres and $t \geq 0$ .

(a)
Show that the drones collide, and find the coordinates of the point of collision.
4 marks
(b)
Find the speed of each drone.
2 marks

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Question 3 · AA-00270Paper 3 · 28 marks

This question asks you to investigate straight lines that lie entirely on a curved surface.

The surface $S$ consists of all points $(x, y, z)$ that satisfy the equation $z = x y$ .

For each $a \in R$ , the line $L_{a}$ has vector equation

r = (\begin{array}{c} a \\ 0 \\ 0 \end{array}) + λ (\begin{array}{c} 0 \\ 1 \\ a \end{array}), λ \in R .

For each $b \in R$ , the line $M_{b}$ has vector equation

r = (\begin{array}{c} 0 \\ b \\ 0 \end{array}) + μ (\begin{array}{c} 1 \\ 0 \\ b \end{array}), μ \in R .

(a)(i)
Consider the line $L_{2}$ .
Write down the coordinates of the point on $L_{2}$ for which $λ = 3$ .
1 mark
(a)(ii)
Verify that this point lies on $S$ .
1 mark
(b)
Show that every point of $L_{a}$ lies on $S$ .
2 marks
(c)
Show that $L_{1}$ and $L_{2}$ are skew.
4 marks
(d)
Show that, for $a \neq c$ , the lines $L_{a}$ and $L_{c}$ are skew.
3 marks
(e)
Show that $L_{a}$ and $M_{b}$ intersect, and find the coordinates of their point of intersection in terms of $a$ and $b$ .
4 marks
(f)
The point $P (p, q, p q)$ lies on $S$ .
Write down the value of $a$ and the value of $b$ for which both $L_{a}$ and $M_{b}$ pass through P.
1 mark
(g)
Let $θ$ be the acute angle between $L_{p}$ and $M_{q}$ .
Show that $\cos θ = \frac{∣ p q ∣}{\sqrt{(1 + p^{2}) (1 + q^{2})}}$ .
3 marks
(h)
Hence show that $L_{p}$ and $M_{q}$ are perpendicular if and only if P lies on the $x$ -axis or on the $y$ -axis.
2 marks
(i)
The plane $Π$ contains both $L_{p}$ and $M_{q}$ .
Show that the Cartesian equation of $Π$ is $q x + p y - z = p q$ .
4 marks
(j)
By substituting $z = x y$ into the equation of $Π$ , show that the only points of $S$ that lie in $Π$ are the points of $L_{p}$ and $M_{q}$ .
3 marks

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Question 4 · AA-00254Paper 1 · 7 marks

The line $L_{1}$ has Cartesian equation

\frac{x - 1}{2} = 3 - y = \frac{z + 2}{4} .

(a)
Find a vector equation of $L_{1}$ .
2 marks
(b)
The line $L_{2}$ has vector equation
$r = (\begin{array}{c} 0 \\ 1 \\ 5 \end{array}) + μ (\begin{array}{c} 3 \\ 1 \\ - 2 \end{array}), μ \in R .$
Show that $L_{1}$ and $L_{2}$ are skew.
5 marks

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Question 5 · AA-00256Paper 2 · 7 marks

Three lines have the following vector equations, where $λ, μ, ν \in R$ .

L_{1} : r = (\begin{array}{c} 1.5 \\ - 2 \\ 0.4 \end{array}) + λ (\begin{array}{c} 2 \\ 0.5 \\ - 1 \end{array})

L_{2} : r = (\begin{array}{c} - 3 \\ 4.1 \\ 2 \end{array}) + μ (\begin{array}{c} - 6 \\ - 1.5 \\ 3 \end{array})

L_{3} : r = (\begin{array}{c} 1.7 \\ - 6.35 \\ - 2.5 \end{array}) + ν (\begin{array}{c} 1.2 \\ 2.5 \\ 0.8 \end{array})

(a)
Determine whether $L_{1}$ and $L_{2}$ are coincident, parallel but distinct, intersecting or skew. Justify your answer.
3 marks
(b)
Show that $L_{1}$ and $L_{3}$ intersect, and find the coordinates of the point of intersection.
4 marks

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Question 6 · AA-00255Paper 1 · 4 marks

The line $L_{1}$ has vector equation

r = (\begin{array}{c} 1 \\ 4 \\ - 2 \end{array}) + λ (\begin{array}{c} 2 \\ - 1 \\ 3 \end{array}), λ \in R .

The line $L_{2}$ passes through the points $P (7, 1, 7)$ and $Q (- 3, 6, - 8)$ .

(a)
Show that the direction of $L_{2}$ is parallel to the direction of $L_{1}$ .
2 marks
(b)
Determine whether $L_{1}$ and $L_{2}$ are coincident. Justify your answer.
2 marks

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