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Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ2.11
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Show that	Question number	11	Adapted from	N/A

Question

Two airplanes, $A$ and $B$ , have position vectors with respect to an origin $O$ given respectively by

$r_{A} = (\begin{matrix} 19 \\ - 1 \\ 1 \end{matrix}) + t (\begin{matrix} - 6 \\ 2 \\ 4 \end{matrix})$

$r_{B} = (\begin{matrix} 1 \\ 0 \\ 12 \end{matrix}) + t (\begin{matrix} 4 \\ 2 \\ - 2 \end{matrix})$

where $t$ represents the time in minutes and $0 \leq t \leq 2.5$ .

Entries in each column vector give the displacement east of $O$ , the displacement north of $O$ and the distance above sea level, all measured in kilometres.

The two airplanes’ lines of flight cross at point $P$ .

Find the three-figure bearing on which airplane $B$ is travelling.

[2]

Show that airplane $A$ travels at a greater speed than airplane $B$ .

[2]

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

[4]

Find the coordinates of $P$ .

[5]

d.i.

Determine the length of time between the first airplane arriving at $P$ and the second airplane arriving at $P$ .

[2]

d.ii.

Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .

[5]

Markscheme

let $ϕ$ be the required angle (bearing)

EITHER

$ϕ = 90 ° - arctan \frac{1}{2} (= arctan 2)$ (M1)

Note: Award M1 for a labelled sketch.

$\cos ϕ = \frac{(\begin{matrix} 0 \\ 1 \end{matrix}) \cdot (\begin{matrix} 4 \\ 2 \end{matrix})}{\sqrt{1} \times \sqrt{20}} (= 0.4472 \dots, = \frac{1}{\sqrt{5}})$ (M1)

$ϕ = arccos (0.4472 \dots)$

THEN

$063 °$ A1

Note: Do not accept $063.6 °$ or $63.4 °$ or $1 . 10^{c}$ .

[2 marks]

METHOD 1

let $|b_{A}|$ be the speed of $A$ and let $|b_{B}|$ be the speed of $B$

attempts to find the speed of one of $A$ or $B$ (M1)

$|b_{A}| = \sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}}$ or $|b_{B}| = \sqrt{4^{2} + 2^{2} + {(- 2)}^{2}}$

Note: Award M0 for $|b_{A}| = \sqrt{19^{2} + {(- 1)}^{2} + 1^{2}}$ and $|b_{B}| = \sqrt{1^{2} + 0^{2} + 12^{2}}$ .

$|b_{A}| = 7.48 \dots (= \sqrt{56})$ (km min^-1) and $|b_{B}| = 4.89 \dots (= \sqrt{24})$ (km min^-1) A1

$|b_{A}| > |b_{B}|$ so $A$ travels at a greater speed than $B$ AG

METHOD 2

attempts to use $speed = \frac{distance}{time}$

${speed}_{A} = \frac{|r_{A} (t_{2}) - r_{A} (t_{1})|}{t_{2} - t_{1}}$ and ${speed}_{B} = \frac{|r_{B} (t_{2}) - r_{B} (t_{1})|}{t_{2} - t_{1}}$ (M1)

for example:

${speed}_{A} = \frac{|r_{A} (1) - r_{A} (0)|}{1}$ and ${speed}_{B} = \frac{|r_{B} (1) - r_{B} (0)|}{1}$

${speed}_{A} = \frac{\sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}}}{1}$ and ${speed}_{B} = \frac{\sqrt{4^{2} + 2^{2} + 2^{2}}}{1}$

${speed}_{A} = 7.48 \dots (2 \sqrt{14})$ and ${speed}_{B} = 4.89 \dots (\sqrt{24})$ A1

${speed}_{A} > {speed}_{B}$ so $A$ travels at a greater speed than $B$ AG

[2 marks]

attempts to use the angle between two direction vectors formula (M1)

$\cos θ = \frac{(- 6) (4) + (2) (2) + (4) (- 2)}{\sqrt{{(- 6)}^{2} + 2^{2} + 4^{2}} \sqrt{4^{2} + 2^{2} + {(- 2)}^{2}}}$ (A1)

$\cos θ = - 0.7637 \dots (= - \frac{7}{\sqrt{84}})$ or $θ = arccos (- 0.7637 \dots) (= 2.4399 \dots)$

attempts to find the acute angle $180 ° - θ$ using their value of $θ$ (M1)

$= 40.2 °$ A1

[4 marks]

for example, sets $r_{A} (t_{1}) = r_{B} (t_{2})$ and forms at least two equations (M1)

$19 - 6 t_{1} = 1 + 4 t_{2}$

$- 1 + 2 t_{1} = 2 t_{2}$

$1 + 4 t_{1} = 12 - 2 t_{2}$

Note: Award M0 for equations involving $t$ only.

EITHER

attempts to solve the system of equations for one of $t_{1}$ or $t_{2}$ (M1)

$t_{1} = 2$ or $t_{2} = \frac{3}{2}$ A1

attempts to solve the system of equations for $t_{1}$ and $t_{2}$ (M1)

$t_{1} = 2$ or $t_{2} = \frac{3}{2}$ A1

THEN

substitutes their $t_{1}$ or $t_{2}$ value into the corresponding $r_{A}$ or $r_{B}$ (M1)

$P (7, 3, 9)$ A1

Note: Accept $\vec{OP} = (\begin{matrix} 7 \\ 3 \\ 9 \end{matrix})$ . Accept $7$ km east of $O$ , $3$ km north of $O$ and $9$ km above sea level.

[5 marks]

d.i.

attempts to find the value of $t_{1} - t_{2}$ (M1)

$t_{1} - t_{2} = 2 - \frac{3}{2}$

$0.5$ minutes ( $30$ seconds) A1

[2 marks]

d.ii.

EITHER

attempts to find $r_{B} - r_{A}$ (M1)

$r_{B} - r_{A} = (\begin{matrix} - 18 \\ 1 \\ 11 \end{matrix}) + t (\begin{matrix} 10 \\ 0 \\ - 6 \end{matrix})$

attempts to find their $D (t)$ (M1)

$D (t) = \sqrt{{(10 t - 18)}^{2} + 1 + {(11 - 6 t)}^{2}}$ A1

attempts to find $r_{A} - r_{B}$ (M1)

$r_{A} - r_{B} = (\begin{matrix} 18 \\ - 1 \\ - 11 \end{matrix}) + t (\begin{matrix} - 10 \\ 0 \\ 6 \end{matrix})$

attempts to find their $D (t)$ (M1)

$D (t) = \sqrt{{(18 - 10 t)}^{2} + {(- 1)}^{2} + {(- 11 + 6 t)}^{2}}$ A1

Note: Award M0M0A0 for expressions using two different time parameters.

THEN

either attempts to find the local minimum point of $D (t)$ or attempts to find the value of $t$ such that $D' (t) = 0$ (or equivalent) (M1)

$t = 1.8088 \dots (= \frac{123}{68})$

$D (t) = 1.01459 \dots$

minimum value of $D (t)$ is $1.01 (= \frac{\sqrt{1190}}{34})$ (km) A1

Note: Award M0 for attempts at the shortest distance between two lines.

[5 marks]

Examiners report

General comment about this question: many candidates were not exposed to this setting of vectors question and were rather lost.

Part (a) Probably the least answered question on the whole paper. Many candidates left it blank, others tried using 3D vectors. Out of those who calculated the angle correctly, only a small percentage were able to provide the correct true bearing as a 3-digit figure.

Part (b) Well done by many candidates who used the direction vectors to calculate and compare the speeds. A number of candidates tried to use the average rate of change but mostly unsuccessfully.

Part (c) Most candidates used the correct vectors and the formula to obtain the obtuse angle. Then only some read the question properly to give the acute angle in degrees, as requested.

Part (d) Well done by many candidates who used two different parameters. They were able to solve and obtain two values for time, the difference in minutes and the correct point of intersection. A number of candidates only had one parameter, thus scoring no marks in part (d) (i). The frequent error in part (d)(ii) was providing incorrect units.

Part (e) Many correct answers were seen with an efficient way of setting the question and using their GDC to obtain the answer, graphically or numerically. Some gave time only instead of actually giving the minimal distance. A number of candidates tried to find the distance between two skew lines ignoring the fact that the lines intersect.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.12—Vector definitions

Show 77 related questions

22M.2.AHL.TZ1.7b:
Given that $|a + b|$ is a minimum, find $p$ .
18M.2.AHL.TZ1.H_11c.iii:
Find the distance between the two submarines at this time.
18M.2.AHL.TZ1.H_11b.ii:
Find the value of t when submarine B passes through P.
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.
17N.1.AHL.TZ0.H_9a.i:
Find, in terms of a and b $\overrightarrow {{\text{OF}}}$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $\Pi$ .
19M.1.AHL.TZ1.H_11d:
Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .
18M.2.AHL.TZ1.H_11c.ii:
Find the value of t when the two submarines are closest together.
19N.1.SL.TZ0.S_9a.i:
$\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}}$ .
19M.1.AHL.TZ1.H_11a.i:
Find how many sets of four points can be selected which can form the vertices of a quadrilateral.
19N.2.SL.TZ0.S_2b:
Write down a direction vector for ${L_3}$ .
19M.1.AHL.TZ1.H_11c:
Write down the value of $\lambda$ corresponding to the point ${\text{A}}$ .
19M.1.AHL.TZ1.H_11a.ii:
Find how many sets of three points can be selected which can form the vertices of a triangle.
18M.2.AHL.TZ1.H_11c.i:
Find an expression for the distance between the two submarines in terms of t.
18M.1.AHL.TZ2.H_9d:
Find the vector equation of the straight line passing through M and normal to the plane $\Pi$ containing ABCD.
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
18M.2.AHL.TZ1.H_11a:
Show that the two submarines would collide at a point P and write down the coordinates of P.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
19M.1.AHL.TZ2.H_2a.i:
Find the vector $\overrightarrow {{\text{AB}}}$ .
19M.1.AHL.TZ2.H_2a.ii:
Find the vector $\overrightarrow {{\text{AC}}}$ .
19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.AHL.TZ1.7a:
Find the possible range of values for $|a + b|$ .
18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
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 ${L_1}$ .
17N.1.AHL.TZ0.H_9b.i:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\lambda$ ;
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.1.SL.TZ2.S_2b:
perpendicular.
18N.2.SL.TZ0.S_8a.i:
Find $\overrightarrow {{\text{AB}}}$ .
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
20N.1.SL.TZ0.S_9b:
Find the value of $m$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
19N.2.SL.TZ0.S_2c:
${L_3}$ passes through the intersection of ${L_1}$ and ${L_2}$ .

Write down a vector equation for ${L_3}$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
17N.1.AHL.TZ0.H_9b.ii:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\mu$ .
19M.1.AHL.TZ2.H_2b:
Hence or otherwise, find the area of the triangle ABC.
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.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
19M.1.AHL.TZ1.H_11e:
Let ${\text{C}}$ be the point on ${l_1}$ with coordinates (1, 0, 1) and ${\text{D}}$ be the point on ${l_2}$ with parameter $\mu = - 2$ .

Find the area of the quadrilateral ${\text{CDBA}}$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
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 $\overrightarrow {AB}$ .
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 $y$ .
19N.2.SL.TZ0.S_2a:
Find the point of intersection of ${L_1}$ and ${L_2}$ .
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.1.AHL.TZ2.H_9f.ii:
Find YZ.
19M.1.AHL.TZ1.H_11b:
Verify that ${\text{P}}$ is the point of intersection of the two lines.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
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.AHL.TZ2.H_9f.i:
Find the coordinates of X, Y and Z.
18M.1.AHL.TZ2.H_9c:
Find the area of the parallelogram ABCD.
17N.1.AHL.TZ0.H_9d:
Deduce an expression for $\overrightarrow {{\text{CD}}}$ in terms of a and b only.
20N.1.SL.TZ0.S_9a:
Express $\vec{AB}$ in terms of $m$ .
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 $L$ .
17N.1.AHL.TZ0.H_9e:
Given that area $\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})$ , find the value of $k$ .
19M.1.SL.TZ2.S_2a:
parallel.
17N.1.AHL.TZ0.H_9c:
Show that $\mu = \frac{1}{{13}}$ , and find the value of $\lambda$ .
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.2.AHL.TZ1.H_11b.i:
Show that submarine B travels in the same direction as originally planned.
17N.1.AHL.TZ0.H_9a.ii:
Find, in terms of a and b $\overrightarrow {{\text{AF}}}$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
20N.1.SL.TZ0.S_9c:
Consider a unit vector $u$ , such that $u = p i - \frac{2}{3} j + \frac{1}{3} k$ , where $p > 0$ .

Point $C$ is such that $\vec{BC} = 9 u$ .

Find the coordinates of $C$ .
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 $ABCD$ .
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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Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line

Topic 3— Geometry and trigonometry

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