RevisionDojo

AHL 3.18—Intersections of lines & planes - IB Questionbank | RevisionDojo

Practice AHL 3.18—Intersections of lines & planes 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 a triangle $XYZ$ such that $X$ has coordinates $(0, 0, 0), Y$ has coordinates $(0, 1, 2)$ and $Z$ has coordinates $(2b, 0, b - 1)$ where $b < 0$

Let $N$ be the midpoint of the line segment $XZ$ .

Find, in terms of $b$ , a Cartesian equation of the plane $\Pi$ containing this triangle.

[5]

Find, in terms of $b$ , the equation of the line $L$ which passes through N and is perpendicular to the plane $П$ .

[2]

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 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 3

HLPaper 1

Let the plane $\Pi_1: 2x - y + 2z = 9$ and the plane $\Pi_2: x + 2y - z = 7$ .

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

[4]

A third plane $\Pi_3: 3x + y + kz = 11$ contains the line $L$ .
Find the value of $k$ .

[3]

Find the acute angle between planes $\Pi_1$ and $\Pi_2$ .

[3]

Determine the equation of the plane perpendicular to both $\Pi_1$ and $\Pi_2$ and passing through the origin.

[3]

Find the coordinates of the point where the line $L$ meets the plane $x+y+z=10$ .

[3]

Question 4

HLPaper 1

Let the line $L$ and plane $\Pi$ be given by
$L:\ \mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+t\begin{pmatrix}1\\2\\2\end{pmatrix},\\ \Pi:\ 2x-y+2z=5.$

Show that $L$ is not contained in $\Pi$ .

[3]

Find the point of intersection of $L$ and $\Pi$ .

[3]

Find the distance from the point $P(5,0,2)$ to the line $L$ .

[4]

Find the shortest distance between the line $L$ and the plane $\Pi_1:\ 2x-y+2z=1.$

[3]

Hence determine the acute angle between line $L$ and plane $\Pi$ .

[3]

Question 5

HLPaper 1

[Maximum Mark: 8] [Without GDC] The point $A(2,5,-1)$ is on the line $L$ , which is perpendicular to the plane with equation $x+y+z=1$

Find the Cartesian equation of the line $L$ .

[3]

Find the point of intersection of the line $L$ and the plane.

[5]

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 straight line, $M_\theta$ , has vector equation $\mathbf{r} = \left( \begin{array}{c} 5 \\ 0 \\ 0 \end{array} \right) + \lambda \left( \begin{array}{c} 5 \\ \sin\theta \\ \cos\theta \end{array} \right)$ , $\lambda$ , $\theta \in \mathbb{R}$ .

The plane $\Sigma_q$ , has equation $x = q$ , $q \in \mathbb{R}$ .

Show that the angle between $M_\theta$ and $\Sigma_q$ is independent of both $\theta$ and $q$ .

[6]

Question 8

HLPaper 1

The points C and D are given by ${\text{C}}(0, 3, -6)$ and ${\text{D}}(6, -5, 11)$ .

The plane Ω is defined by the equation $4x - 3y + 2z = 20$ .

Find a vector equation of the line $M$ passing through the points C and D.

[3]

Find the coordinates of the point of intersection of the line $M$ with the plane $\Omega$ .

[3]

Question 9

HLPaper 1

[Maximum Mark : 7] [Without GDC] Find the coordinates of the point of intersection of the line $L$ with the plane $P$ where:

$L: \frac{x+3}{2}=\frac{y-1}{-1}=\frac{z-1}{2}$ and $P: 2 x+3 y-z=-5$

[7]

Question 10

HLPaper 2

The points $A(5, -2, 5)$ , $B(5, 4, -1)$ , $C(-1, -2, -1)$ and $D(7, -4, -3)$ are the vertices of a right-pyramid.The line $L$ passes through the point $D$ and is perpendicular to plane $\Phi$

Find the vectors ${\vec{AB}}$ and ${\vec{AC}}$ .

[2]

Show that the Cartesian equation of the plane $\Phi$ that contains the triangle $ABC$ is $-x+y+z=-2$ .

[3]

Find a vector equation of the line $L$ .

[1]

Hence determine the minimum distance, $d_{\text{min}}$ , from $D$ to $\Phi$ .

[4]

Use a vector method to show that $B\hat{A}C = 60^\circ$ .

[3]

Find the volume of right-pyramid ${ABCD}$ .

[4]

AHL 3.18—Intersections of lines & planes Questionbank