AHL 3.15—Classification of lines Questionbank

Practice AHL 3.15—Classification of lines with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The motion of a particle is given by the equation $\vec{r}=\begin{pmatrix} 3 \\ 5 \end{pmatrix}+t\begin{pmatrix} 4 \\ 3 \end{pmatrix}$ , where $t$ is in seconds.

Find initial position of the particle and its position after 2 seconds.

[6]

Find the velocity and the speed of the particle.

[3]

How far from the origin is the particle after 2 seconds?

[4]

How far from its initial position is the particle after 2 seconds?

[4]

Question 2

HLPaper 1

Consider the plane $\Pi_1$ given by the equation $x + 2y - z = 4$ and the plane $\Pi_2$ given by the equation $3x - y + 4z = 10$ .

Find a vector equation of the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify that the line of intersection found is perpendicular to the normal vectors of both planes.

[3]

Question 3

HLPaper 1

Two lines in three-dimensional space are given by $L_1:\ \mathbf r = \begin{pmatrix} 0\\ 1\\ 2 \end{pmatrix} +\lambda \begin{pmatrix} 1\\ -1\\ 2 \end{pmatrix}, \qquad L_2:\ \mathbf r = \begin{pmatrix} 2\\ 3\\ 0 \end{pmatrix} +\mu \begin{pmatrix} 2\\ -2\\ 4 \end{pmatrix}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[4]

Find the acute angle between the lines.

[2]

Let $S=(1,0,0)$ . Find the shortest distance from $S$ to line $L_1$ , and the coordinates of the foot $F$ of the perpendicular.

[5]

Let $P=(2,3,0)$ , a point on $L_2$ . Find all points $Q$ on $L_2$ such that $\overrightarrow{PQ}$ makes an angle of $90^\circ$ with the direction of $L_1$ .

[4]

Question 4

HLPaper 1

Two lines in three-dimensional space are given by $L_1: \mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2: \mathbf{r}=\begin{pmatrix}5\\0\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix},\qquad \lambda,\mu\in\mathbb{R}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[3]

Find the acute angle $\theta$ between $L_1$ and $L_2$ , giving an exact expression and a decimal value (degrees) to 1 d.p.

[3]

Let $S = (2, 0, 5)$ . Find the shortest distance from $S$ to the line $L_1$ , and the coordinates of the foot $F$ of the perpendicular from $S$ to $L_1$ .

[4]

Find all points $Q$ on $L_2$ such that the vector $\overrightarrow{PQ}$ makes an angle of $45^\circ$ with the direction of $L_1$ , where $P$ is the intersection point from part (b). Give exact parameter values $\mu$ and coordinates of $Q$ if they exist.

[5]

Question 5

HLPaper 1

Consider two lines in three-dimensional space, given by their vector equations: $L_1:\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ and $L_2:\mathbf{r}_2 = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix} + s \begin{pmatrix} a \\ 1 \\ 2 \end{pmatrix}$

Find the values of $a$ for which the lines are skew.

[7]

Question 6

HLPaper 1

The lines $l_1$ and $l_2$ have the following vector equations where $\lambda, \mu \in \mathbb{R}$ and $m \in \mathbb{R}$ .

$l_1: \mathbf{r}_1 = \begin{pmatrix} 3 \\ -2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ m \end{pmatrix}$

$l_2: \mathbf{r}_2 = \begin{pmatrix} -1 \\ -4 \\ -2m \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -5 \\ -m \end{pmatrix}$

The plane $\Pi$ has Cartesian equation $x + 4y - z = p$ where $p \in \mathbb{R}$ .

Show that $l_1$ and $l_2$ are never perpendicular to each other.

[3]

Given that $l_1$ and $\Pi$ have no points in common, find the value of $m$ .

[2]

Find the condition on the value of $p$ .

[2]

Question 7

HLPaper 1

A boat moves with constant velocity along a straight line. Its velocity vector is given by $v=\binom{4}{3}$ . At time $t=0$ it is at the point $(-2,1)$ .

Find the speed of the boat.

[3]

Write down a vector equation representing the position of the boat, giving your answer in the form $r=a+t b$ .

[2]

Find the coordinates of the boat when $t=2$ .

[4]

Question 8

HLPaper 1

Consider the line $L_1$ given by the vector equation $\mathbf{r} = \begin{pmatrix} 8 \\ 4 \\ 9 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ 9 \end{pmatrix}$ and the plane $\pi$ given by the equation $2x - y + 3z = 12$ .

Find the point of intersection between the line $L_1$ and the plane $\pi$ .

[3]

Determine if the line $L_1$ is parallel, perpendicular, or neither to the plane $\pi$ .

[4]

Question 9

HLPaper 1

Consider the line $L_1$ given by the equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\pi$ given by the equation $2x - y + z = 5$ .

Find the point of intersection of the line $L_1$ and the plane $\pi$ .

[4]

Determine if the line $L_1$ is parallel, perpendicular, or neither to the normal of the plane $\pi$ .

[4]

Question 10

HLPaper 1

Two lines in three-dimensional space are given by $L_1:\ \mathbf{r} = \begin{pmatrix} 1\\ -1\\ 2 \end{pmatrix} +\lambda \begin{pmatrix} 2\\ 1\\ -1 \end{pmatrix}, \qquad L_2:\ \mathbf{r} = \begin{pmatrix} 5\\ 1\\ 0 \end{pmatrix} +\mu \begin{pmatrix} -2\\ -1\\ 1 \end{pmatrix}, \quad \lambda,\mu\in\mathbb{R}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[4]

Find the acute angle $\theta$ between $L_1$ and $L_2$ , giving an exact expression and a decimal value (degrees) to 1 d.p.

[3]

Let $S=(0,0,0)$ . Find the shortest distance from $S$ to the line $L_1$ , and the coordinates of the foot $F$ of the perpendicular from $S$ to $L_1$ .

[5]

Let $P=(5,1,0)$ , a point on both lines. Find all points $Q$ on $L_2$ such that the vector $\overrightarrow{PQ}$ makes an angle of $60^\circ$ with the direction of $L_1$ . Justify your answer.

[4]