Wednesday, December 12, 2018

Vector

$\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 $\D \mathbf{i}$ 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. [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)
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 km h$\D^{-1}$ 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]

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

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