CIE Vector (Additional Mathematics -2018)

$\def\lvec#1{\overrightarrow{#1}}\def\D{\displaystyle}$

$\newcommand{\iixi}[2]{\left(\begin{array}{c}#1\\#2\end{array}\right)}$

1 (CIE 2012, s, paper 12, question 12either)

EITHER

At 12 00 hours, a ship has position vector $\D 54\mathbf{i}+16$ km relative to a lighthouse, where $\D \mathbf{i}$ isa unit vector due East and $\D\mathbf{j}$ is a unit vector due North. The ship is travelling with a speed of 20 km $\D h^{-1}$ in the direction $\D 3\mathbf{i} +4\mathbf{j}.$

(i) Show that the position vector of the ship at 15 00 hours is $\D(90\mathbf{i} + 64\mathbf{j})$ km.[2]

(ii) Find the position vector of the ship t hours after 12 00 hours. [2]

A speedboat leaves the lighthouse at 14 00 hours and travels in a straight line to intercept the ship. Given that the speedboat intercepts the ship at 16 00 hours, find

(iii) the speed of the speedboat, [3]

(iv) the velocity of the speedboat relative to the ship, [1]

(v) the angle the direction of the speedboat makes with North. [2]

2 (CIE 2012, s, paper 12, question 12or) 

The position vectors of points $\D A$ and $\D B$ relative to an origin $\D O$ are $\D a$ and $\D b$ respectively. The point $\D P$ is such that $\D \lvec{OP} = 54\lvec{OB}$ . The point $\D Q$ is such that $\D \lvec{AQ} = 13\lvec{AB}$. The point $\D R$ lies on $\D OA$ such that $\D RQP$ is a straight line where $\D \lvec{OR} =\lambda\lvec{OA}$ and $\D \lvec{QR} = \mu \lvec{PR} .$

(i) Express $\D \lvec{OQ}$ and $\D\lvec{PQ}$ in terms of $\D\mathbf{a}$ and $\D\mathbf{b}.$ [2]

(ii) Express $\lvec{QR}$ in terms of $\D \lambda, \mathbf{a}$ and $\D \mathbf{b}.$ [2]

(iii) Express $\D \lvec{QR}$ in terms of $\D\mu , \mathbf{a}$ and $\D \mathbf{b}.$ [3]

(iv) Hence find the value of $\D\lambda$ and of $\D \mu.$ [3]

3 (CIE 2012, s, paper 21, question 8)

Relative to an origin O, the position vectors of the points A and B are $\D 2\mathbf{i} – 3\mathbf{j}$ and 11i + 42j respectively.

(i) Write down an expression for $\D \lvec{AB.}$ [2]

The point C lies on AB such that $\D \lvec{AC} = \frac{1}{3}\lvec{AB}.$

(ii) Find the length of $D\lvec{OC.}$ [4]

The point $\D D$ lies on $\D \lvec{OA}$ such that $\D \lvec{DC}$  is parallel to $\D \lvec{OB.}$

(iii) Find the position vector of $\D D.$ [2]

4 (CIE 2012, w, paper 12, question 1)

It is given that $\D\mathbf{a} = \iixi 43, \mathbf{b} = \iixi{-1}{2}$ and $\D \mathbf{c} = \iixi{21}{2}.$

(i) Find $\D |\mathbf{a} + \mathbf{b} + \mathbf{c}|.$ [2]

(ii) Find λ and μ such that  $\D\lambda \mathbf{a}+ \mu \mathbf{b} = \mathbf{c}.$ [3]

5 (CIE 2012, w, paper 13, question 5)

A pilot flies his plane directly from a point A to a point B, a distance of 450 km. The bearing of B from A is 030°. A wind of 80 km h$\D^{-1}$ is blowing from the east. Given that the plane can travel at 320 km h$\D^{-1}$ in still air, find

(i) the bearing on which the plane must be steered, [4]

(ii) the time taken to fly from A to B. [4]

6 (CIE 2012, w, paper 21, question 7)

In this question $\D\iixi 10$ is a unit vector due east and $\D\iixi 01$ is a unit vector due north. At 12 00 a coastguard, at point O, observes a ship with position vector $\D\iixi{16}{12}$ km relative to O. The ship is moving at a steady speed of 10kmh$\D^{-1}$ on a bearing of 330°.

(i) Find the value of p such that $\D\iixi{-5}{p}$ kmh$\D^{-1}$ represents the velocity of the ship. [2]

(ii) Write down, in terms of $\D t,$ the position vector of the ship, relative to $\D O, t$ hours after 12 00. [2]

(iii) Find the time when the ship is due north of O. [2]

(iv) Find the distance of the ship from $\D O$ at this time. [2]

7 (CIE 2012, w, paper 22, question 9)

A plane, whose speed in still air is 420 km h$\D^{-1},$ travels directly from $\D A$ to $\D B,$ a distance of 1000 km. The bearing of $\D B$ from $\D A$ is 230° and there is a wind of $80 \mbox{km h}^{-1}$ from the east.

(i) Find the bearing on which the plane was steered. [4]

(ii) Find the time taken for the journey. [4]

8 (CIE 2012, w, paper 23, question 4)

The points $\D X, Y$ and $\D Z$ are such that $\D\lvec{XY} = 3\lvec{YZ} .$ The position vectors of $\D X$ and $\D Z,$ relative to an origin $\D O,$ are $\D\iixi{4}{-27}$ and $\D\iixi{20}{-7}$ respectively. Find the unit vector in the direction $\D\lvec{OY}.$ [5]

9 (CIE 2013, s, paper 11, question 9)

The figure shows points $\D A, B$ and $\D C$ with position vectors $\mathbf{a,b}$ and $\D \mathbf{c}$ respectively, relative to an origin $\D O.$ The point $\D P$ lies on $\D AB$ such that $\D AP:AB = 3:4.$ The point $\D Q$ lies on $\D OC$ such that $\D OQ:QC = 2:3.$

(i) Express $\D\lvec{AP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ and hence show that $\D\lvec{OP} =\frac{1}{4}(\mathbf{a} +3\mathbf{b}).$ [3]

(ii) Find $\D\lvec{PQ}$ in terms of $\D\mathbf{a,b}$ and $\D\mathbf{c}.$ [3]

(iii) Given that $\D 5\lvec{PQ} = 6\lvec{BC},$ find $\D\mathbf{c}$ in terms of $\D\mathbf{a}$ and $\D\mathbf{b}.$ [2]

10 (CIE 2013, s, paper 21, question 10)

A plane, whose speed in still air is 240 kmh$\D^{-1},$ flies directly from $\D A$ to $\D B,$ where $\D B$ is 500 km from $\D A$ on a bearing of 032°. There is a constant wind of 50 kmh$\D^{-1}$ blowing from the west.

(i) Find the bearing on which the plane is steered. [4]

(ii) Find, to the nearest minute, the time taken for the flight. [4]

11 (CIE 2013, s, paper 22, question 4)

The position vectors of the points $\D A$ and $\D B,$ relative to an origin $\D O,$ are $\D 4\mathbf{i}- 21\mathbf{j}$ and $\D 22\mathbf{i} - 30\mathbf{j}$ respectively. The point $\D C$ lies on $\D AB$ such that $\D \lvec{AB} = 3\lvec{AC.}$

(i) Find the position vector of $\D C$ relative to $\D O.$ [4]

(ii) Find the unit vector in the direction $\lvec{OC.}$ [2]

***************

12 CIE $2013, \mathrm{w}$, paper 21, question 9$)$

The diagram shows a river with parallel banks. The river is $40 \mathrm{~m}$ wide and is flowing with speed of $1.8 \mathrm{~ms}^{-1} .$ A canoe travels in a straight line from a point $P$ on one bank to a point $Q$ on the opposite bank $70 \mathrm{~m}$ downstream from $P .$ Given that the canoe takes $12 \mathrm{~s}$ to travel from $P$ to $Q$, calculate the speed of the canoe in still water and the angle to the bank that the canoe was steered. $[8]$

$13 \mathrm{CIE} 2013, \mathrm{w}$, paper 23, question 11)

In this question $\mathrm{i}$ is a unit vector due east and $\mathrm{j}$ is a unit vector due north. At time $t=0$ boat $A$ leaves the origin $O$ and travels with velocity $(2 \mathbf{i}+4 \mathrm{j}) \mathrm{kmh}^{-1}$. Also at time $t=0$ boat $B$ leaves the point with position vector $(-21 \mathrm{i}+22 \mathrm{j}) \mathrm{km}$ and travels with velocity $(5 \mathbf{i}+3 \mathrm{j}) \mathrm{kmh}^{-1}$

(i) Write down the position vectors of boats $A$ and $B$ after $t$ hours.

(ii) Show that $A$ and $B$ are $25 \mathrm{~km}$ apart when $t=2$. $[3]$

(iii) Find the length of time for which $A$ and $B$ are less than $25 \mathrm{~km}$ apart.

14 (CIE $2014, \mathrm{~s}$, paper 11, question 2 )

Vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a}=\left(\begin{array}{l}4 \\ 3\end{array}\right), \mathbf{b}=\left(\begin{array}{l}2 \\ 2\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}-5 \\ 2\end{array}\right)$

(i) Show that $|\mathbf{a}|=|\mathbf{b}+\mathbf{c}|$.

(ii) Given that $\lambda \mathrm{a}+\mu \mathrm{b}=7 \mathrm{c}$, find the value of $\lambda$ and of $\mu$.

15 (CIE $2014, \mathrm{~s}$, paper 12, question 10 )

In this question $\mathrm{i}$ is a unit vector due East and $\mathrm{j}$ is a unit vector due North. At 1200 hours, a ship leaves a port $P$ and travels with a speed of $26 \mathrm{kmh}^{-1}$ in the direction $5 \mathbf{i}+12 \mathrm{j}$.

(i) Show that the velocity of the ship is $(10 \mathbf{i}+24 \mathbf{j}) \mathrm{kmh}^{-1}$.

(ii) Write down the position vector of the ship, relative to $P$, at 1600 hours.

(iii) Find the position vector of the ship, relative to $P, t$ hours after 1600 hours. $[2]$

At 1600 hours, a speedboat leaves a lighthouse which has position vector $(120 \mathrm{i}+81 \mathrm{j}) \mathrm{km}$, relative to $P$, to intercept the ship. The speedboat has a velocity of $(-22 \mathbf{i}+30 \mathbf{j}) \mathrm{kmh}^{-1}$.

(iv) Find the position vector, relative to $P$, of the speedboat $t$ hours after 1600 hours. $[1]$

(v) Find the time at which the speedboat intercepts the ship and the position vector, relative to $P$, of the point of interception. $[4]$

16 (CIE $2014, \mathrm{~s}$, paper 23 , question 11)

In the diagram $\overline{O A}=2 \mathbf{a}$ and $\overline{O B}=5 \mathbf{b}$. The point $M$ is the midpoint of $O A$ and the point $N$ lies on $O B$ such that $O N: N B=3: 2$.

(i) Find an expression for the vector $\overrightarrow{M B}$ in terms of a and $\mathbf{b}$.

The point $P$ lies on $A N$ such that $\overrightarrow{A P}=\lambda \overrightarrow{A N}$.

(ii) Find an expression for the vector $\overrightarrow{A P}$ in terms of $\lambda_{1}$ a and $\mathbf{b}$. $[2]$

(iii) Find an expression for the vector $\overrightarrow{M P}$ in terms of $\lambda, \mathbf{a}$ and $\mathbf{b}$. $[2]$

(iv) Given that $M, P$ and $B$ are collinear, find the value of $\lambda$ $[4$

17 (CIE $2014, \mathrm{w}$, paper 11, question 12 )

The position vectors of points $A$ and $B$ relative to an origin $O$ are a and $\mathrm{b}$ respectively. The point $P$ is such that $\overrightarrow{O P}=\mu \overrightarrow{O A}$. The point $Q$ is such that $\overrightarrow{O Q}=\lambda \overrightarrow{O B}$. The lines $A Q$ and $B P$ intersect at the point $R$

(i) Express $\overrightarrow{A Q}$ in terms of $\lambda, \mathbf{a}$ and $\mathbf{b}$. $[1]$

(ii) Express $\overrightarrow{B P}$ in terms of $\mu, \mathbf{a}$ and $\mathbf{b}$. $[1]$

It is given that $3 \overrightarrow{A R}=\overrightarrow{A Q}$ and $8 \overrightarrow{B R}=7 \overrightarrow{B P}$

(iii) Express $\overline{O R}$ in terms of $\lambda, \mathrm{a}$ and $\mathrm{b}$.

(iv) Express $\overrightarrow{O R}$ in terms of $\mu, \mathrm{a}$ and $\mathrm{b}$.

(v) Hence find the value of $\mu$ and of $\lambda$. $[3]$

$18 \mathrm{CIE} 2014, \mathrm{w}$, paper 23 , question 3)

Points $A$ and $B$ have coordinates $(-2,10)$ and $(4,2)$ respectively. $C$ is the mid-point of the line $A B$. Point $D$ is such that $\overrightarrow{C D}=\left(\begin{array}{c}12 \\ 9\end{array}\right)$.

(i) Find the coordinates of $C$ and of $D$. [3]

(ii) Show that $C D$ is perpendicular to $A B$. [3]

(iii) Find the area of triangle $A B D$. [2]

19 (CIE $2014, \mathrm{w}$, paper 23 , question 5)

In the diagram $\overrightarrow{O P}=\mathbf{b}, \overrightarrow{P Q}=\mathbf{a}$ and $\overrightarrow{O R}=3 \mathbf{a}$. The lines $O Q$ and $P R$ intersect at $X$.

(i) Given that $\overline{O X}=\mu \bar{O} \dot{Q}$, express $\overrightarrow{O X}$ in terms of $\mu, \mathbf{a}$ and $\mathbf{b}$.[1]

(ii) Given that $\overrightarrow{R X}=\lambda \overrightarrow{R P}$, express $\overrightarrow{O X}$ in terms of $\lambda_{\text {, }} \mathbf{a}$ and $\mathbf{b}$. [2]

(iii) Hence find the value of $u$ and of $\lambda$ and state the value of the ratio $\frac{R X}{X P}$. [3]

20 (CIE 2015 , s, paper 12, question 7)

In the diagram $\overrightarrow{A B}=4 \mathrm{a}, \overrightarrow{B C}=\mathbf{b}$ and $\overrightarrow{D C}=7 \mathbf{a}$. The lines $A C$ and $D B$ intersect at the point $X$. Find, in terms of $\mathbf{a}$ and $\mathbf{b}$.

(i) $\overrightarrow{D A}$

(ii) $\overrightarrow{D B}$

Given that $\overrightarrow{A X}=\lambda \overrightarrow{A C}$, find, in terms of $\mathbf{a}, \mathbf{b}$ and $\lambda$,

(iii) $\overrightarrow{A \dot{X}}$

(iv) $\overline{D X}$.

Given that $\overrightarrow{D X}=\mu \overrightarrow{D B}$

(v) find the value of $\lambda$ and of $\mu$.

21 (CIE 2015, s, paper 21 , question 7)

(a) The four points $O, A, B$ and $C$ are such that $\overrightarrow{O A}=5 \mathbf{a}, \quad \overrightarrow{O B}=15 \mathbf{b}, \quad \overrightarrow{O C}=24 \mathbf{b}-3 \mathbf{a}$. Show that $B$ lies on the line $A C$.

(b) Relative to an origin $O$, the position vector of the point $P$ is $1-4 j$ and the position vector of the point $Q$ is $3 \mathbf{i}+7 \mathbf{j}$. Find

(i) $|\overrightarrow{P Q}|$, $[2]$

(ii) the unit vector in the direction $\overrightarrow{P Q}$, $[1]$

(iii) the position vector of $M$, the mid-point of $P Q$. $[2]$

22 (CIE 2015, s, paper 22, question 4)

A river, which is $80 \mathrm{~m}$ wide, flows at $2 \mathrm{~ms}^{-1}$ between parallel, straight banks. A man wants to row his boat straight across the river and land on the other bank directly opposite his starting point. He is able to row his boat in still water at $3 \mathrm{~ms}^{-1}$. Find

(i) the direction in which he must row his boat, $[2]$

(ii) the time it takes him to cross the river. [3]

$23(\mathrm{CIE} 2015, \mathrm{w}$, paper 21, question 3)

Relative to an origin $O$, points $A, B$ and $C$ have position vectors $\left(\begin{array}{c}5 \\ 4\end{array}\right),\left(\begin{array}{r}-10 \\ 12\end{array}\right)$ and $\left(\begin{array}{r}6 \\ -18\end{array}\right)$ respectively. All distances are measured in kilometres. A man drives at a constant speed directly from $A$ to $B$ in 20 minutes.

(i) Calculate the speed in $\mathrm{kmh}^{-1}$ at which the man drives from $A$ to $B$. [3]

He now drives directly from $B$ to $C$ at the same speed.

(ii) Find how long it takes him to drive from $B$ to $C$.

24 (CIE $2015, \mathrm{w}$, paper 23, question 12 )

A plane that can travel at $250 \mathrm{kmh}^{-1}$ in still air sets oft on a bearing of $070^{\circ} .$ A wind with speed $w \mathrm{kmh}^{-1}$ from the south blows the plane off course so that the plane actually travels on a bearing of $060^{\circ}$.

Find, in $\mathrm{kmh}^{-1}$, the resultant speed $V$ of the plane and the windspeed $w$.

25 (CIE 2016, march, paper 22, question 10)

(a) The vectors $\mathbf{p}$ and $\mathbf{q}$ are such that $\mathbf{p}=11 \mathbf{i}-24 \mathbf{j}$ and $\mathbf{q}=2 \mathbf{i}+\alpha \mathbf{j}$.

(i) Find the value of each of the constants $\alpha$ and $\beta$ such that $\mathbf{p}+2 \mathbf{q}=(\alpha+\beta) \mathbf{i}-20 \mathbf{j}$. $[3]$

(ii) Using the values of $\alpha$ and $\beta$ found in part (i), find the unit vector in the direction $\mathbf{p}+2 \mathbf{q}$. [2]

(b) The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ with respect to an origin $O$. The point $C$ lies on $A B$ and is such that $A B: A C$ is $1: \lambda$. Find an expression for $\overrightarrow{O C}$ in terms of $\mathbf{a}, \mathbf{b}$ and $\lambda$. $\quad[3]$

(c) The points $S$ and $T$ have position vectors $\mathrm{s}$ and $\mathbf{t}$ with respect to an origin $O$. The points $O, S$ and $T$ do not lie in a straight line. Given that the vector $2 \mathrm{~s}+\mu \mathrm{t}$ is parallel to the vector $(\mu+3) \mathrm{s}+9 \mathrm{t}$, where $\mu$ is a positive constant, find the value of $\mu .$

26 (CIE $2016, \mathrm{~s}$, paper 11, question 7)

The diagram shows a river with parallel banks. The river is $75 \mathrm{~m}$ wide and is flowing with a speed of $2.4 \mathrm{~ms}^{-1}$. A speedboat travels in a straight line from a point $A$ on one bank to a point $B$ on the opposite bank, $30 \mathrm{~m}$ downstream from $A$. The speedboat can travel at a speed of $4.5 \mathrm{~ms}^{-1}$ in still water.

(i) Find the angle to the bank and the direction in which the speedboat is steered. $[4]$

(ii) Find the time the speedboat takes to travel from $A$ to $B$. $[4]$

27 (CIE $2016, \mathrm{~s}$, paper 12, question 3) Vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{e}$ are such that $\mathbf{a}=\left(\begin{array}{l}2 \\ y\end{array}\right), \mathbf{b}=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{r}-5 \\ 5\end{array}\right)$

(i) Given that $|\mathbf{a}|=|\mathbf{b}-\mathbf{c}|$, find the possible values of $y$. $[3$

(ii) Given that $\mu(\mathbf{b}+\mathbf{c})+4(\mathbf{b}-\mathbf{c})=\lambda(2 \mathbf{b}-\mathbf{c})$, find the value of $\mu$ and of $\lambda$.

28 (CIE $2016, \mathrm{~s}$, paper 21 , question 7) $O, P, Q$ and $R$ are four points such that $\overrightarrow{O P}=\mathbf{p}, \overrightarrow{O Q}=\mathbf{q}$ and $\overrightarrow{O R}=3 \mathbf{q}-2 \mathbf{p}$

(i) Find, in terms of $\mathbf{p}$ and $\mathbf{q}$,

(a) $\overrightarrow{P Q}$ $[1]$

(b) $\overrightarrow{Q R}$.

(ii) Justifying your answer, what can be said about the positions of the points $P, Q$ and $R$ ? $[2]$

(iii) Given that $\overrightarrow{O P}=\mathbf{i}+3 \mathbf{j}$ and that $\overrightarrow{O Q}=2 \mathbf{i}+\mathbf{j}$, find the unit vector in the direction $\overrightarrow{O R}$. [3]

29 CIE $2016, \mathrm{w}$, paper 21, question 10$)$

The town of Cambley is $5 \mathrm{~km}$ east and $p \mathrm{~km}$ north of Edwintown so that the position vector of Cambley from Edwintown is $\left(\begin{array}{c}5000 \\ 1000 p\end{array}\right)$ metres. Manjit sets out from Edwintown at the same time as Raj sets out from Cambley. Manjit sets out from Edwintown on a bearing of $020^{\circ}$ at a speed of $2.5 \mathrm{~ms}^{-1}$ so that her position vector relative to Edwintown after $t$ seconds is given by $\left(\begin{array}{l}2.5 t \cos 70^{\circ} \\ 2.5 t \cos 20^{\circ}\end{array}\right)$ metres. Raj sets out from Cambley on a bearing of $310^{\circ}$ at $2 \mathrm{~ms}^{-1}$.

(i) Find the position vector of Raj relative to Edwintown after $t$ seconds. $[2]$

Manjit and Raj mect after $T$ seconds.

(ii) Find the value of $T$ and of $p$. $[5]$

30 (CIE $2016, \mathrm{w}$, paper 23 , question 9)

In this question $\mathrm{i}$ is a unit vector due cast and $\mathbf{j}$ is a unit vector due north. Units of length and velocity are metres and metres per second respectively.

The initial position vectors of particles $A$ and $B$, relative to a fixed point $O$, are $\mathrm{i}+5 \mathrm{j}$ and $q \mathrm{i}-15 \mathrm{j}$ respectively, $A$ and $B$ start moving at the same time. $A$ moves with velocity $p \mathbf{i}-3 \mathbf{j}$ and $B$ moves with velocity $3 \mathbf{i}-\mathbf{j}$.

(i) Given that $A$ travels with a speed of $5 \mathrm{~ms}^{-1}$, find the value of the positive constant $p$.

(ii) Find the direction of motion of $B$ as a bearing correct to the nearest degree. [2]

(iii) State the position vector of $A$ after $t$ seconds. [1]

(iv) State the position vector of $B$ after $t$ seconds.

(v) Find the time taken until $A$ and $B$ meet. [2]

(vi) Find the position vector of the point where $A$ and $B$ mect. [1]

(vii) Find the value of the constant $q$.

31 (CIE 2017 , march, paper 12, question 7)

(a) A vector $\mathbf{v}$ has a magnitude of 102 units and has the same direction as $\left[\begin{array}{r}8 \\ -15\end{array}\right)$. Find $\mathbf{v}$ in the form $\left(\begin{array}{l}a \\ b\end{array}\right)$, where $a$ and $b$ are integers.

(b) Vectors $\mathrm{c}=\left(\begin{array}{l}4 \\ 3\end{array}\right)$ and $\mathrm{d}=\left(\begin{array}{c}p-q \\ 5 p+q\end{array}\right)$ are such that $\mathbf{c}+2 \mathbf{d}=\left(\begin{array}{l}p^{2} \\ 27\end{array}\right)$. Find the possible values of the constants $p$ and $q . \quad[6]$

32 (CIE 2017, march, paper 22, question 7)

(a) Calculate the magnitude and bearing of the resultant velocity of $10 \mathrm{~ms}^{-1}$ on a bearing of $240^{\circ}$ and $5 \mathrm{~ms}^{-1}$ due south.

(b) A car travelling east along a road at a velocity of $38 \mathrm{kmh}^{-1}$ passes a lorry travelling west on the same road at a velocity of $56 \mathrm{kmh}^{-1}$, Write down the velocity of the lorry relative to the car. $[2]$

33 CIE 2017, s, paper 11, question 5 )

(a) The diagram shows a figure $O A B C$, where $\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $\overrightarrow{O C}=\mathbf{c}$. The lines $A C$ and $O B$ intersect at the point $M$ where $M$ is the midpoint of the line $A C$.

(i) Find, in terms of a and $c$, the vector $\overrightarrow{O M}$.

(ii) Given that $O M: M B=2: 3$, find $\mathbf{b}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

(b) Vectors $\mathbf{i}$ and $\mathbf{j}$ are unit vectors parallel to the $x$ -axis and $y$ -axis respectively.

The vector $\mathbf{p}$ has a magnitude of 39 units and has the same direction as $-10 \mathbf{i}+24 \mathrm{j}$.

(i) Find $\mathbf{p}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.

(ii) Find the vector $q$ such that $2 p+q$ is parallel to the positive $y$ -axis and has a magnitude of 12 units. $[3$

(iii) Hence show that $|\mathbf{q}|=k \sqrt{5}$, where $k$ is an integer to be found. $[2]$

34 (CIE 2017, s, paper 12, question 3) Vectors $\mathbf{i}$ and $\mathbf{j}$ are unit vectors parallel to the $x$ -axis and $y$ -axis respectively.

(a) The vector $\mathbf{v}$ has a magnitude of $3 \sqrt{5}$ units and has the same direction as $\mathbf{i}-2 \mathbf{j}$. Find $\mathbf{v}$ givin your answer in the form $a \mathbf{i}+b \mathbf{j}$, where $a$ and $b$ are integers.

(b) The velocity vector $w$ makes an angle of $30^{\circ}$ with the positive $x$ -axis and is such that $|\mathbf{w}|=2$ Find $\mathbf{w}$ giving your answer in the form $\sqrt{c} \mathbf{i}+d \mathbf{j}$, where $c$ and $d$ are integers.

35 (CIE 2017, s, paper 23, question 4)

(a) Vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a}=\left(\begin{array}{r}5 \\ -6\end{array}\right), \mathbf{b}=\left(\begin{array}{r}11 \\ -15\end{array}\right)$ and $3 \mathbf{a}+\mathbf{c}=\mathbf{b}$.

(i) Find $\mathbf{c}$.

(ii) Find the unit vector in the direction of $\mathbf{b}$.

(b) In the diagram, $\overrightarrow{O P}=\mathbf{p}$ and $\overrightarrow{O Q}=\mathbf{q} .$ The point $R$ lies on $P Q$ such that $P R=3 R Q .$ Find $\overrightarrow{O R}$ in terms of $\mathbf{p}$ and $\mathbf{q}$, simplifying your answer.

36 (CIE $2017, \mathrm{w}$, paper 12 , question 8 )

The diagram shows a river which is $120 \mathrm{~m}$ wide and is flowing at $4 \mathrm{~ms}^{-1}$. Points $A$ and $B$ are on opposite sides of the river such that $B$ is $50 \mathrm{~m}$ downstream from $A$. A man needs to cross the river from $A$ to $B$ in a boat which can travel at $5 \mathrm{~ms}^{-1}$ in still water.

(i) Show that the man must point his boat upstream at an angle of approximately $65^{\circ}$ to the bank.

(ii) Find the time the man takes to cross the river from $A$ to $B$.

37 (CIE $2017, \mathrm{w}$, paper 21, question 10) In this question $\mathrm{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north. Units of length and velocity are metres and metres per second respectively.

The initial position vectors of particles $A$ and $B$, relative to a fixed point $O$, are $2 \mathbf{i}+4 \mathbf{j}$ and $10 \mathbf{i}+14 \mathbf{j}$ respectively. Particles $A$ and $B$ start moving at the same time. $A$ moves with constant velocity $\mathbf{i}+\mathbf{j}$ and $B$ moves with constant velocity $-2 \mathbf{i}-3 \mathbf{j}$. Find

(i) the position vector of $A$ after $t$ seconds,

(ii) the position vector of $B$ after $t$ seconds.

It is given that $X$ is the distance between $A$ and $B$ after $t$ seconds.

(iii) Show that $X^{2}=(8-3 t)^{2}+(10-4 t)^{2}$.

(iv) Find the value of $t$ for which $(8-3 t)^{2}+(10-4 t)^{2}$ has a stationary value and the corresponding value of $X$

38 CIE $2017, \mathrm{w}$, paper 23, question 5 )

The diagram shows points $O, A, B, C, D$ and $X .$ The position vectors of $A, B$ and $C$ relative to $O$ are $\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $\overrightarrow{O C}=\frac{3}{2} \mathbf{b} .$ The vector $\overrightarrow{C D}=3 \mathbf{a}$.

(i) If $\overrightarrow{O X}=\lambda \overrightarrow{O D}$ express $\overrightarrow{O X}$ in terms of $\lambda, \mathbf{a}$ and $\mathbf{b}$. $[1]$

(ii) If $\overrightarrow{A X}=\mu \overrightarrow{A B}$ express $\overrightarrow{O X}$ in terms of $\mu, \mathbf{a}$ and $\mathbf{b}$.

(iii) Use your two expressions for $\overline{O X}$ to find the value of $\lambda$ and of $\mu$. $[3]$

(iv) Find the ratio $\frac{A X}{X B}$. $[1]$

(v) Find the ratio $\frac{O X}{X D}$. $[1]$

$39 \quad($ CIE $2017, \mathrm{w}$, paper 23, question 8$)$

A man, who can row a boat at $3 \mathrm{~ms}^{-1}$ in still water, wants to cross a river from $A$ to $B$ as shown in the diagram. $A B$ is perpendicular to both banks of the river. The river, which is $50 \mathrm{~m}$ wide, is flowing at $1 \mathrm{~ms}^{-1}$ in the direction shown. The man points his boat at an angle $a^{\circ}$ to the bank. Find

(i) the angle $\alpha$, $[2]$

(ii) the resultant speed of the boat from $A$ to $B$, $[2]$

(iii) the time taken for the boat to travel from $A$ to $B$. $[2]$

On another occasion the man points the boat in the same direction but the river speed has increased to $1.8 \mathrm{~ms}^{-1}$ and as a result he lands at the point $C$.

(iv) State the time taken for the boat to travel from $A$ to $C$ and hence find the distance $B C .$ $[2]$

40 (CIE 2018, march, paper 12, question 6)

The diagram shows the quadrilateral $O A B C$ such that $\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}$ and $\overrightarrow{O C}=\mathbf{c}$. It is given that $A M: M C=2: 1$ and $O M: M B=3: 2 .$

(i) Find $\overrightarrow{A C}$ in terms of $\mathbf{a}$ and $\mathbf{c}$. $[1]$

(ii) Find $\overline{O M}$ in terms of $\mathbf{a}$ and $\mathbf{c}$. $[2]$

(iii) Find $\overline{O M}$ in terms of $\mathbf{b}$.

(iv) Find $5 \mathrm{a}+10 \mathrm{c}$ in terms of $\mathbf{b}$.

(v) Find $\overrightarrow{A B}$ in terms of a and $\mathbf{c}$, giving your answer in its simplest form. $[2]$

41 (CIE 2018, march, paper 22, question 5)

A river is 104 metres wide and the current flows at $0.5 \mathrm{~ms}^{-1}$ parallel to its banks. A woman can swim at $1.6 \mathrm{~ms}^{-1}$ in still water. She swims from point $A$ and aims for point $B$ which is directly opposite, but she is carried downstream to point $C$. Calculate the time it takes the woman to swim across the river and the distance downstream, $B C$, that she travels.

42 (CIE 2018, s, paper 11, question 8)

(a) Given that $\mathbf{p}=2 \mathbf{i}-5 \mathbf{j}$ and $\mathbf{q}=\mathbf{i}-3 \mathbf{j}$, find the unit vector in the direction of $3 \mathbf{p}-4 \mathbf{q}$.

(b) A river flows between parallel banks at a speed of $1.25 \mathrm{kmh}^{-1}$. A boy standing at point $A$ on one bank sends a toy boat across the river to his father standing directly opposite at point $B$. The toy boat, which can travel at $v \mathrm{kmh}^{-1}$ in still water, crosses the river with resultant speed $2.73 \mathrm{kmh}^{-1}$ along the line $A B$.

(i) Calculate the value of $v$. $[2]$

The direction in which the boy points the boat makes an angle $\theta$ with the line $A B$.

(ii) Find the value of $\theta$.

43 (CIE $2018, \mathrm{~s}$, paper 22 , question 7) 

Vectors $\mathbf{i}$ and $\mathbf{j}$ are vectors parallel to the $x$ -axis and $y$ -axis respectively. Given that $\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}, \mathbf{b}=\mathbf{i}-5 \mathbf{j}$ and $\mathbf{c}=3 \mathbf{i}+11 \mathbf{j}$, find

(i) the exact value of $|\mathbf{a}+\mathbf{c}|$

(ii) the value of the constant $m$ such that $\mathbf{a}+m \mathbf{b}$ is parallel to $\mathbf{j}$,

(iii) the value of the constant $n$ such that $n \mathbf{a}-\mathbf{b}=\mathbf{c}$.

Answers

1. (ii)$\D 54\mathbf{i} + 16\mathbf{j} + (12\mathbf{i} + 16\mathbf{j})t$

(iii) 64.8

(iv) $\D 39\mathbf{i}+24\mathbf{j}$ (v)51.9

2. (i) $\D\frac{2}{3}\mathbf{a}-\frac{11}{12}\mathbf{b}$

(ii) $\D\lambda\mathbf{a}- (2/3)\mathbf{a}- (1/3)\mathbf{b}$

(iii) $\D\frac{\mu}{1-\mu}(\frac{2}{3}\mathbf{a}-\frac{11}{12}\mathbf{b})$

(iv)$\D \mu=\frac{4}{15},\lambda=\frac{10}{11}$ 3. (i) $\D 9\mathbf{i}+45\mathbf{j}$ (ii)13

(iii) $\D\frac{4}{3}\mathbf{i}-2\mathbf{j}$

4. (i)25

(ii)$\D\lambda = 4,\mu=-5$

5. (i)Bearing 043;(ii) 1.65

6. (i) $\D 5\sqrt{3}$

(ii) $\D\frac{16-5t}{12+8.66t}$

(iii)1512 (iv)39.7

7. (i)bearing 223

(ii)2h 5min

8. $\D\frac{1}{5}\iixi{4}{-3}$

9. (i) $\D \frac{1}{4}(\mathbf{a}+3\mathbf{b})$

(ii) $\D\frac{2}{5}\mathbf{c}-\frac{1}{4}\mathbf{a}-\frac{3}{4}\mathbf{b}$

(iii) $\D \mathbf{c} = \frac{9\mathbf{b}-5\mathbf{a}}{16}$

10. (i) 022(ii)1h54m

11. (i)10i-24j

(ii) $\D \frac{1}{13}(5\mathbf{i}-12\mathbf{j})$

12. $\alpha=39.6, v=5.23$

13. (ai) $(2 i+4 j) t$ $-21 i+22 j+(5 i+3 j) t$

(iii) 13 hours

14. (ii) $\lambda=49, \mu=80.5$

15. (ii) $40 \mathrm{i}+96 \mathrm{j}$

(iii) $(40 \mathrm{i}+96 \mathrm{j})+(10 \mathrm{i}+24 \mathrm{j}) \mathrm{t}$

(iv) $120 \mathrm{i}+81 \mathrm{j}+(-22 \mathrm{i}+30 \mathrm{j}) \mathrm{t}$

(v) $18: 30,65 \mathrm{i}+156 \mathrm{j}$

16. (i) $5 \mathrm{~b}-\mathrm{a}$

(ii) $\lambda(3 b-2 a)$

(iii) $a+\lambda(3 b-2 a)$

$($ iv $) \lambda=5 / 7$

17. (i) $\lambda b-a$

(ii) $\mu a-b$

(iii) $(2 / 3) a+\lambda / 3 b$

$(\mathrm{iv})(1 / 8) b+(7 \mu / 8 a)$

$(\mathrm{v}) \lambda=3 / 8, \mu=16 / 21$

18. (i)C $(1,6), D(13,15)$

(iii) $A=75$

19. (i) $\mu(a+b)$

(ii) $3 a+\lambda(b-3 a)$

(iii) $\mu=\lambda=.75, \frac{R X}{X P}=3$

20. (i) $3 \mathrm{a}-\mathrm{b}$ (ii) $7 \mathrm{a}-\mathrm{b}$

(iii) $\lambda(4 a+b)$

(iv) $3 a-b+\lambda(4 a+b)$

$(\mathrm{v}) \lambda=4 / 11, \mu=7 / 11$

21. (bi) $5 \sqrt{5}$

(ii) $\frac{1}{5 \sqrt{5}}(2 i+11 j)$

(iii) $2 i+1.5 j$

22. (i) $48.2,131.8$ (ii) $35.8$

23. (i) $51 \mathrm{~km} / \mathrm{h}$, (ii) $40 \mathrm{~min}$

24. $V=271 \mathrm{~km} / \mathrm{h}, w=50.1 \mathrm{~km} / \mathrm{h}$

25. (ai) $\alpha=2, \beta=13$

(ii) $\frac{15 i-20 j}{25}$

(b) $(1-\lambda) a+\lambda b$

(c) $\mu=3$

26. (i) Direction is $82.1$ to the bank, upstream.

(ii) $16.8$

27. (i) $y=\pm 6$

(ii) $\mu=4 / 3, \lambda=8 / 3$

28. (ia)q-p (b) $2 \mathrm{q}-2 \mathrm{p}$

(ii) collinear

(iii) $(1 / 5)(4 i-3 j)$

29. (i) $r_{j}=\left(\begin{array}{c}5000 \\ 1000 p\end{array}\right)+\left(\begin{array}{c}-2 \cos 40 \\ 2 \cos 50\end{array}\right) t$

(ii) $p=2.23$

30. (i) $p=4$ (ii) 108

(iii) $r_{A}=\left(\begin{array}{l}1 \\ 5\end{array}\right)+t\left(\begin{array}{c}4 \\ -3\end{array}\right)$

(iv) $r_{B}=\left(\begin{array}{c}q \\ -15\end{array}\right)+t\left(\begin{array}{c}3 \\ -1\end{array}\right)$

(v) 10 (vi) $\left(\begin{array}{c}41 \\ -25\end{array}\right)$ (vi) 11

31. (a) $\left(\begin{array}{c}48 \\ -90\end{array}\right)$

(b) $p=2, q=2 ; p=10, q=-38$

32. (a) $13.2,220.9$

(b) $94 \mathrm{~km} / \mathrm{h}$,west

33. (ai)(a+c)/2

(aii) $(5 / 4)(\mathrm{a}+\mathrm{c})$

(bi)p $=-15 i+36 \mathrm{j}$

(bii) $\mathrm{q}=30 \mathrm{i}-60 \mathrm{j}$

(biii) $30 \sqrt{5}$

34. (a) 3i-6j

(b) $\sqrt{3} i+6 j$

35. (ai) $\left(\begin{array}{c}-4 \\ 3\end{array}\right)$

(aii) $\frac{1}{\sqrt{346}}\left(\begin{array}{c}11 \\ -15\end{array}\right)$

(b)p $/ 4+3 \mathrm{q} / 4$

36. (ii) $26.5$

37. (i) $r_{A}=(2 i+4 j)+t(i+j)$

(ii) $r_{B}=(10 i+14 j)+t(2 i 3 j)$

(iv) $0.4$

38. (i) $\lambda(1.5 b+3 a)$

(ii) $a+\mu(b-a)$

(iii) $\mu=1 / 3, \lambda=2 / 9$

(iv) $1 / 2(\mathrm{v}) 2 / 7$

39. (i) $70.5(\mathrm{ii}) 2.83$

(iii) $17.7$ (iv) $14.1$

40. (i)c $c-a$

(ii) $(2 / 3) c+(1 / 3) a$

(iii) $(3 / 5) b$

(iv) $9 c$

$(\mathrm{v})(-4 / 9) a+(10 / 9) c$

41. $32.5$

42. Unit vector $=\frac{2 i-3 j}{\sqrt{13}}$

$v=3$, Angle to $A B=24.6$

43. (i) $\sqrt{221}$

(ii) $m=-2$

(iii) $n=2$