AHL 3.14—Vector equation of line Questionbank

Practice AHL 3.14—Vector equation of line 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

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 3

HLPaper 1

Find the equation of the line passing through $A(1,3,5)$ and parallel to:

$x$ -axis

[2]

$y-$ axis

[2]

$z$ -axis

[2]

Question 4

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 5

HLPaper 1

Express the following $3 D$ lines in the form $r=a+\lambda b$

$L_{1}: \frac{x-2}{3}=\frac{y+4}{7}=\frac{z-1}{5}$

[1]

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

[1]

$L_{3}: \frac{x-2}{3}=\frac{y+4}{7}=z=10$

[1]

$L_{4}: \frac{x-2}{3}=\frac{y+4}{0}=\frac{z+1}{1}$

[1]

Question 6

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 7

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 8

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 9

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 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]