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AHL 3.15—Classification of lines - IB Questionbank | RevisionDojo

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

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 value of $a$ for which the lines are skew.

[7]

Question 2

HLPaper 1

The motion of a particle is given by the equation $\vec{r}=\binom{3}{5}+t\binom{4}{3}$ , 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 3

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 the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify if the line of intersection found is perpendicular to the normal of either plane.

[3]

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 $Q$

[5]

Question 5

HLPaper 1

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

Show that $L_1$ and $L_2$ are skew (not parallel and do not intersect).

[3]

Find the acute angle $\theta$ between $L_1$ and $L_2$ in exact form.

[3]

Find points $P\in L_1$ and $Q\in L_2$ such that $\overrightarrow{PQ}$ is perpendicular to both lines. (That is, the common perpendicular segment.)
Give $P,Q$ in exact coordinates.

[5]

Hence find the shortest distance between $L_1$ and $L_2$ in exact surd form.

[2]

Question 6

HLPaper 1

The paths $l_1$ 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 surface $\Pi$ has Cartesian equation $x + 4y - z = p$ where $p \in \mathbb{R}$ .

Given that $l_1$ and $\Pi$ have no points in common, find

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

[3]

the value of $m$ .

[2]

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

Consider a line $L$ and a plane $P$ in a three-dimensional space.

The line $L$ is given by the parametric equations $x = 2 + t$ , $y = 3 - 2t$ , $z = 1 + 4t$ . The plane $P$ is given by the equation $2x - y + z = 5$ . Find the point of intersection of the line $L$ and the plane $P$ .

[3]

Determine whether the line $L$ is parallel, perpendicular, or neither to the normal vector of the plane $P$ .

[3]

AHL 3.15—Classification of lines Questionbank