IB Questionbank

User interface language: English | Español

Date	May 2021	Marks available	5	Reference code	21M.1.AHL.TZ2.8
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Find	Question number	8	Adapted from	N/A

Question

The lines $l_{1}$ and $l_{2}$ have the following vector equations where $λ, μ \in ℝ$ .

$l_{1} : r_{1} = (\begin{matrix} 3 \\ 2 \\ - 1 \end{matrix}) + λ (\begin{matrix} 2 \\ - 2 \\ 2 \end{matrix})$

$l_{2} : r_{2} = (\begin{matrix} 2 \\ 0 \\ 4 \end{matrix}) + μ (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$

Show that $l_{1}$ and $l_{2}$ do not intersect.

[3]

Find the minimum distance between $l_{1}$ and $l_{2}$ .

[5]

Markscheme

METHOD 1

setting at least two components of $l_{1}$ and $l_{2}$ equal M1

$3 + 2 λ = 2 + μ (1)$

$2 - 2 λ = - μ (2)$

$- 1 + 2 λ = 4 + μ (3)$

attempt to solve two of the equations eg. adding $(1)$ and $(2)$ M1

gives a contradiction (no solution), eg $5 = 2$ R1

so $l_{1}$ and $l_{2}$ do not intersect AG

Note: For an error within the equations award M0M1R0.
Note: The contradiction must be correct to award the R1.

METHOD 2

$l_{1}$ and $l_{2}$ are parallel, so $l_{1}$ and $l_{2}$ are either identical or distinct. R1

Attempt to subtract two position vectors from each line,

e.g. $(\begin{matrix} 3 \\ 2 \\ - 1 \end{matrix}) - (\begin{matrix} 2 \\ 0 \\ 4 \end{matrix}) (= (\begin{matrix} 1 \\ 2 \\ - 5 \end{matrix}))$ M1

$(\begin{matrix} 3 \\ 2 \\ - 1 \end{matrix}) \neq k (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$ A1

[3 marks]

METHOD 1

$l_{1}$ and $l_{2}$ are parallel (as $(\begin{matrix} 2 \\ - 2 \\ 2 \end{matrix})$ is a multiple of $(\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$ )

let $A$ be $(3, 2, - 1)$ on $l_{1}$ and let $B$ be $(2, 0, 4)$ on $l_{2}$

Attempt to find vector $\vec{AB} (= (\begin{matrix} - 1 \\ - 2 \\ 5 \end{matrix}))$ (M1)

Distance required is $\frac{|v \times \vec{AB}|}{|v|}$ M1

$= \frac{1}{\sqrt{3}} |(\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}) \times (\begin{matrix} - 1 \\ - 2 \\ 5 \end{matrix})|$ (A1)

$= \frac{1}{\sqrt{3}} |(\begin{matrix} 3 \\ 6 \\ 3 \end{matrix})|$ A1

minimum distance is $\sqrt{18} (= 3 \sqrt{2})$ A1

METHOD 2

$l_{1}$ and $l_{2}$ are parallel (as $(\begin{matrix} 2 \\ - 2 \\ 2 \end{matrix})$ is a multiple of $(\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$ )

let $A$ be a fixed point on $l_{1}$ eg $(3, 2, - 1)$ and let $B$ be a general point on $l_{2}$ $(2 + μ, - μ, 4 + μ)$

attempt to find vector $\vec{AB}$ (M1)

$\vec{AB} = (\begin{matrix} - 1 \\ - 2 \\ 5 \end{matrix}) + μ (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}) (μ \in ℝ)$ A1

$|\vec{AB}| = \sqrt{{(- 1 + μ)}^{2} + {(- 2 - μ)}^{2} + {(5 + μ)}^{2}} (= \sqrt{3 μ^{2} + 12 μ + 30})$ M1

EITHER

null A1

$|\vec{AB}| = \sqrt{3 {(μ + 2)}^{2} + 18}$ to obtain $μ = - 2$ A1

THEN

minimum distance is $\sqrt{18} (= 3 \sqrt{2})$ A1

METHOD 3

let $A$ be $(3, 2, - 1)$ on $l_{1}$ and let $B$ be $(2 + μ, - μ, 4 + μ)$ on $l_{2}$ (M1)

(or let $A$ be $(2, 0, 4)$ on $l_{2}$ and let $B$ be $(3 + 2 λ, 2 - 2 λ, - 1 + 2 λ)$ on $l_{1}$ )

$\vec{AB} = (\begin{matrix} - 1 \\ - 2 \\ 5 \end{matrix}) + μ (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}) (μ \in ℝ)$ (or $\vec{AB} = (\begin{matrix} 2 λ + 1 \\ - 2 λ + 2 \\ 2 λ - 5 \end{matrix})$ ) A1

$(\begin{matrix} μ - 1 \\ - μ - 2 \\ μ + 5 \end{matrix}) \cdot (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}) = 0$ (or $(\begin{matrix} 2 λ + 1 \\ - 2 λ + 2 \\ 2 λ - 5 \end{matrix}) \cdot (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}) = 0$ ) M1

$μ = - 2$ or $λ = 1$ A1

minimum distance is $\sqrt{18} (= 3 \sqrt{2})$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.13—Scalar (dot) product

Show 77 related questions

22M.2.AHL.TZ2.11c:
Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.
18M.1.SL.TZ1.S_9d:
Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.
17M.1.SL.TZ1.S_8d.ii:
Hence or otherwise, find one point on ${L_2}$ which is $\sqrt 5$ units from C.
19N.1.SL.TZ0.S_9a.i:
$\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}}$ .
18M.1.SL.TZ1.S_6:
Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).
17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π $_3$ .
17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
17M.1.AHL.TZ1.H_5a:
Find the area of the parallelogram ABCD.
18N.1.SL.TZ0.S_5:
Consider the vectors a = $\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)$ .

Find the possible values of p for which a and b are parallel.
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
17M.2.AHL.TZ2.H_9d:
Show that the line (BC) lies in the plane Π $_1$ .
16N.1.SL.TZ0.S_4a:
Find a vector equation of the line that passes through P and Q.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
17N.1.SL.TZ0.S_9a.i:
Show that $\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)$ .
18M.2.SL.TZ2.S_8c:
Find the area of triangle PQR.
16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .
18N.1.AHL.TZ0.H_5a:
Find and simplify an expression for a • b in terms of $t$ .
17N.1.SL.TZ0.S_9b:
Find the value of $p$ .
18M.1.SL.TZ2.S_1a:
Find a vector equation for L₁.
19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.AHL.TZ1.7c:
Find $q$ such that $| q | = | b |$ and $q$ is perpendicular to $a$ .
18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
22M.2.AHL.TZ2.11e:
Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .
18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
17M.1.SL.TZ1.S_8a.ii:
Hence, write down a vector equation for ${L_1}$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.2.SL.TZ2.S_7:
The vector equation of line $L$ is given by r $= \left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 4 \\ 5 \\ { - 1} \end{array}} \right)$ .

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.
19M.1.SL.TZ2.S_2b:
perpendicular.
18N.2.SL.TZ0.S_8a.i:
Find $\overrightarrow {{\text{AB}}}$ .
17M.2.AHL.TZ2.H_9c:
Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
18M.1.AHL.TZ2.H_1:
The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.
17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
19N.1.SL.TZ0.S_9b:
Given that ${\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}$ , show that $k = 7$ .
18M.2.SL.TZ2.S_8a.ii:
Find $\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
17N.1.SL.TZ0.S_9a.ii:
Find a vector equation for $L$ .
21M.1.AHL.TZ2.5:
Given any two non-zero vectors, $a$ and $b$ , show that ${|a \times b|}^{2} = {|a|}^{2} {|b|}^{2} - {(a \cdot b)}^{2}$ .
18M.2.SL.TZ2.S_8b:
Find the angle between PQ and PR.
17N.2.SL.TZ0.S_3a:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B \hat{A} C = 60 °$ .
17M.1.SL.TZ1.S_8a.i:
Find $\overrightarrow {AB}$ .
19M.1.AHL.TZ1.H_1:
Let a = $\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)$ , $k \in \mathbb{R}$ .

Given that a and b are perpendicular, find the possible values of $k$ .
18M.1.SL.TZ1.S_9a:
Show that $\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)$
18N.2.SL.TZ0.S_8b.i:
Find the value of $y$ .
18M.2.SL.TZ2.S_8a.i:
Find $\mathop {{\text{PQ}}}\limits^ \to$ .
17N.1.SL.TZ0.S_9c:
The point D has coordinates $({q^2},{\text{ }}0,{\text{ }}q)$ . Given that $\overrightarrow {{\text{DC}}}$ is perpendicular to $L$ , find the possible values of $q$ .
21N.2.AHL.TZ0.11a.i:
Find the vector $\vec{AB}$ and the vector $\vec{AC}$ .
21N.2.AHL.TZ0.11a.ii:
Hence find the equation of $Π_{1}$ , expressing your answer in the form $a x + b y + c z = d$ , where $a, b, c, d \in ℤ$ .
21N.2.AHL.TZ0.11d.i:
Find the reflection of the point $B$ in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11d.ii:
Hence find the vector equation of the line formed when $L$ is reflected in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11c.i:
Show that at the point $P, λ = \frac{3}{4}$ .
21N.2.AHL.TZ0.11c.ii:
Hence find the coordinates of $P$ .
18M.2.SL.TZ2.S_8d:
Hence or otherwise find the shortest distance from R to the line through P and Q.
17M.1.SL.TZ2.S_2b:
Given that c = a + 2b, find c.
17M.1.SL.TZ2.S_2a:
Find the value of $k$ .
19M.1.SL.TZ1.S_6:
The magnitudes of two vectors, u and v, are 4 and $\sqrt 3$ respectively. The angle between u and v is $\frac{\pi }{6}$ .

Let w = u − v. Find the magnitude of w.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
17M.2.AHL.TZ2.H_9g:
Find the acute angle between the planes Π $_2$ and Π $_3$ .
18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
17N.2.SL.TZ0.S_3b:
Let $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)$ . Find ${\rm{B\hat AC}}$ .
18M.1.SL.TZ2.S_1b:
The vector $\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)$ is perpendicular to $\mathop {{\text{AB}}}\limits^ \to$ . Find the value of p.
EXN.2.AHL.TZ0.11c:
Show that the Cartesian equation of the plane $Π$ that contains the triangle $ABC$ is $- x + y + z = - 2$ .
EXN.2.AHL.TZ0.11d.i:
Find a vector equation of the line $L$ .
18N.1.AHL.TZ0.H_5b:
Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .
17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
19M.1.SL.TZ2.S_2a:
parallel.
19N.1.SL.TZ0.S_9c:
Calculate the area of triangle ${\text{AOC}}$ .
17M.1.SL.TZ1.S_8c:
The lines ${L_1}$ and ${L_1}$ intersect at $C(9,{\text{ }}13,{\text{ }}z)$ . Find $z$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
17M.1.AHL.TZ1.H_5b:
By using a suitable scalar product of two vectors, determine whether ${\rm{A\hat BC}}$ is acute or obtuse.
EXN.1.AHL.TZ0.8a:
Show that $l_{1}$ and $l_{2}$ are never perpendicular to each other.
17M.1.SL.TZ1.S_8b:
A second line ${L_2}$ , has equation r = $\left( {\begin{array}{*{20}{c}} 1 \\ {13} \\ { - 14} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right)$ .

Given that ${L_1}$ and ${L_2}$ are perpendicular, show that $p = 2$ .
EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid $ABCD$ .
21M.1.AHL.TZ2.8a:
Show that $l_{1}$ and $l_{2}$ do not intersect.
21N.2.AHL.TZ0.11b:
The line $L$ is the intersection of $Π_{1}$ and $Π_{2}$ . Verify that the vector equation of $L$ can be written as $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

Hide related questions

Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line

Topic 3— Geometry and trigonometry » AHL 3.15—Classification of lines

Topic 3— Geometry and trigonometry » AHL 3.16—Vector product

Topic 3— Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

View options