### Subscribe Us # Vector (CIE) (Kinematics)

$\def\lvec#1{\overrightarrow{#1}}\def\D{\displaystyle}$
$\newcommand{\iixi}{\left(\begin{array}{c}#1\\#2\end{array}\right)}$

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 $\D \mathbf{i}$ is
a 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. 
(ii) Find the position vector of the ship t hours after 12 00 hours. 
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, 
(iv) the velocity of the speedboat relative to the ship, 
(v) the angle the direction of the speedboat makes with North. 

2 (CIE 2012, s, paper 12, question 12or)
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 $\D\mathbf{b}.$ 
(ii) Express $\lvec{QR}$ in terms of $\D \lambda, \mathbf{a}$ and $\D \mathbf{b}.$ 
(iii) Express $\D \lvec{QR}$ in terms of $\D\mu , \mathbf{a}$ and $\D \mathbf{b}.$ 
(iv) Hence find the value of $\D\lambda$ and of $\D \mu.$ 

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
$\D \lvec{AB.}$ 
The point C lies on AB such that
$\D \lvec{AC} = \frac{1}{3}\lvec{AB}.$
(ii) Find the length of $D\lvec{OC.}$ 
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 $\D D.$ 

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

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, 
(ii) the time taken to fly from A to B. 

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

(iii) Find the time when the ship is due north of O. 
(iv) Find the distance of the ship from $\D O$ at this time. 

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 km h$\D^{-1}$ from the east.
(i) Find the bearing on which the plane was steered. 
(ii) Find the time taken for the journey. 

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 $\D\lvec{OY}.$ 

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 $\D\lvec{OP} =\frac{1}{4}(\mathbf{a} +3\mathbf{b}).$ 
(ii) Find $\D\lvec{PQ}$ in terms of $\D\mathbf{a,b}$ and $\D\mathbf{c}.$ 
(iii) Given that $\D 5\lvec{PQ} = 6\lvec{BC},$ find $\D\mathbf{c}$ in terms of $\D\mathbf{a}$ and $\D\mathbf{b}.$ 

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

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 $\D O.$ 
(ii) Find the unit vector in the direction $\lvec{OC.}$ 

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