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Date	May 2021	Marks available	3	Reference code	21M.1.AHL.TZ2.8
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Show that	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

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Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.
18M.1.SL.TZ1.S_9d:
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17M.1.SL.TZ1.S_8d.ii:
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18M.1.SL.TZ1.S_6:
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This is shown in the following diagram.

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Find the area of the parallelogram ABCD.
18N.1.SL.TZ0.S_5:
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17M.2.AHL.TZ2.H_9d:
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16N.1.SL.TZ0.S_4a:
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18N.2.SL.TZ0.S_8d:
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16N.1.SL.TZ0.S_4b:
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18N.1.AHL.TZ0.H_5a:
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17N.1.SL.TZ0.S_9b:
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18M.1.SL.TZ2.S_1a:
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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:
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22M.2.AHL.TZ2.11e:
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Find the minimum value of $D (t)$ .
18M.1.SL.TZ1.S_9b.i:
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17M.1.SL.TZ1.S_8a.ii:
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EXN.2.AHL.TZ0.11d.ii:
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19M.2.SL.TZ2.S_7:
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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:
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18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
18M.1.AHL.TZ2.H_1:
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17M.2.AHL.TZ2.H_9b:
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19N.1.SL.TZ0.S_9b:
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EXN.2.AHL.TZ0.11a:
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17M.1.SL.TZ1.S_8a.i:
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19M.1.AHL.TZ1.H_1:
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18N.2.SL.TZ0.S_8b.i:
Find the value of $y$ .
18M.2.SL.TZ2.S_8a.i:
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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 $q$ .
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21N.2.AHL.TZ0.11a.ii:
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Find the reflection of the point $B$ in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11d.ii:
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21N.2.AHL.TZ0.11c.i:
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21N.2.AHL.TZ0.11c.ii:
Hence find the coordinates of $P$ .
18M.2.SL.TZ2.S_8d:
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17M.1.SL.TZ2.S_2b:
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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.TZ2.8b:
Find the minimum distance between $l_{1}$ and $l_{2}$ .
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 $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

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Markscheme

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