AHL 3.18—Intersections of lines & planes Questionbank

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 $\Pi$ .

[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 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

[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 4

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 5

HLPaper 1

A straight line, $M_\theta$ , has vector equation $\mathbf{r} = \begin{pmatrix} 5 \\ 0 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 5 \\ \sin\theta \\ \cos\theta \end{pmatrix}$ , $\lambda, \theta \in \mathbb{R}$ .

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 6

HLPaper 1

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

The plane $\Omega$ 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 7

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 8

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 $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[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]

Question 9

HLPaper 1

Consider the points $A(2,-1,0), B(3,0,1), C(1,-1,2)$

Determine the Cartesian equation of the plane $A B C$ .

[5]

Find the area of the triangle $A B C$ .

[5]

The line $L$ is perpendicular to plane $A B C$ and passes through $A$ . Find a vector equation of $L$ .

[3]

The point $D(6,-7,2)$ lies on $L$ . Find the volume of the pyramid $A B C D$ .

[5]

Question 10

HLPaper 1

A line $M$ passes through points $X(-3, 4, 2)$ and $Y(-1, 3, 3)$ . The line $M$ also passes through the point $Z(3, 1, p)$ .

Show that $\overrightarrow{\text{XY}} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$

[1]

Find a vector equation for $M$ .

[2]

Find the value of $p$ .

[5]

The point $W$ has coordinates $(q^2, 0, q)$ . Given that $\overrightarrow{WZ}$ is perpendicular to $M$ , find the possible values of $q$ .

[7]