A military helicopter flies from an airbase at A to a battlefield atj1p ,j ajny/0,ysk(a*pregd5 B and then to a *aejkay,015pn,j jd(y/ spgr hospital at H . The routes taken by the helicopter are given by the vectors
$\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}
-140 \\
70
\end{array}\right) \text { and } \overrightarrow{\mathrm{BH}}=\left(\begin{array}{c}
165 \\
-20
\end{array}\right) \text {. }$
Distances are measured in kilometres.
1. Determine the vector $ \overrightarrow{\mathrm{AH}}$ .
After evacuating wounded soldiers from the battlefield to the front-line hospital, the helicopter returns back to the airbase.$\begin{pmatrix}
a \\
b
\end{pmatrix}$ a =    b =   
2. Write down the vector that describes this return flight.$\begin{pmatrix}
a \\
b
\end{pmatrix}$ a =    b =   
3. Find the direct distance from the hospital to the airbase. ≈    km

  A golf buggy drives in a straight pathm4 lgm/rj8 3v*z :e no.tj7.vfet+d yc*ef*mr on a wide fairway. The buggy's path is given by the parametric equations x=2t+11 and y=3t−5, where t is the drive ti3 * :7ornm d4jt/et vj*z* mlem8.fr.f+y ecgvme in seconds (t≥0) and distances are measured in meters.
1.(1)Write down the initial position of the buggy.(a,b) a=   b=  
(2)Find the position of the buggy after 30 seconds.B(a,b) a=   b=  
2.(1)Write down the velocity vector of the buggy.$\begin{pmatrix}
a \\
b
\end{pmatrix}$ a =    b =   
(2)Find the speed of the buggy. ≈    $ms^{-1}$

  A camel walks in a straight line on flat grassland. The camel's path is given bhgk j.5obdnqym4cb z21/j erb */.dw8y the parametric equations x=3 t-2 and y=1+4 t , where t is the walk time, in hours, t \geq 0 . Distances are measured in kilometres/gq/ bj dwmhron1e*b zc4.5. yj28kbd.
1. (1)Write down the initial position of the camel.(a,b) a=   b=  
(2)Find the position of the camel after 2 hours. B (a,b) a=   b=  
2. (1) Write down the velocity vector of the camel.$\begin{pmatrix}
a \\
b
\end{pmatrix}$ a =    b =   
(2)Find the speed of the camel.    $kmh^{-1}$

  The displacement of a remote-controlled toy yacht is modelled by the vectl/1obsrtil2:drkv l36e t +z9 or equasib136 oelvr l:9tztkr +2l/d tion
$\mathbf{r}=\left(\begin{array}{c}
-5 \\
12
\end{array}\right)+t\left(\begin{array}{c}
1.5 \\
-2
\end{array}\right),$
where t is the sail time, in seconds, $ t \geq 0$ , and distances are measured in metres.
Ali is standing on a boat dock at $\mathrm{O}(0,0)$ and controls the yacht.
1. Write down the position of the yacht at :
(1). t=0 ;(a,b) a=   b=  
(2) t=3 .(a,b) a=   b=  
2. Determine the speed the yacht is travelling.    $ms^{-1}$
3. Determine :
(1) the distance between Ali and the yacht at t=0 ;    m
(2) an expression for the distance between Ali and the yacht for t \geq 0 .
4. Hence, or otherwise, determine :
(1) the minimum distance of the yacht from Ali;    m
(2) the time when this occurs. ≈    seconds

In this question, distances are m.gv .5:g0m vj wwlebv.easured in kilometres.
Two boats, $\mathrm{A} and \mathrm{B}$ , are observed from an origin $\mathrm{O} . Relative to \mathrm{O} $, their position vectors at time t hours after midday are given by
$\begin{array}{l}
\mathbf{r}_{\mathrm{A}}=\left(\begin{array}{c}
-4 \\
3
\end{array}\right)+t\left(\begin{array}{l}
4 \\
3
\end{array}\right) \\
\mathbf{r}_{\mathrm{B}}=\left(\begin{array}{c}
-2 \\
9
\end{array}\right)+t\left(\begin{array}{l}
5 \\
0
\end{array}\right)
\end{array}$
Find the minimum distance between the two boats.

In this question, distances are measured +vj,prrj-tqg9k )e*x6a n ba0g:s y 4win kilometres.
Two tankers, A and B, are observed from an origin O. Relative to O, their position vectors at time t hours after 6:00pm are given by
$\begin{array}{l}
\mathbf{r}_{\mathrm{A}}=\left(\begin{array}{l}
1 \\
1
\end{array}\right)+t\left(\begin{array}{l}
7 \\
4
\end{array}\right) \\
\mathbf{r}_{\mathrm{B}}=\left(\begin{array}{l}
6 \\
9
\end{array}\right)+t\left(\begin{array}{l}
5 \\
3
\end{array}\right)
\end{array}$
Find the time when the distance between the two tankers is at a minimum.

  In this question, distance is in metres. Jack and John are wvy3cn cbef*u ,a1+8mflying airplanes in a straight line at a constant speed. Jack's airplane passey8 *,e+3ncvfb1muacw s through a point $\mathrm{P}$ . Its position, t seconds after it passes through $\mathrm{P}$ , is given by $\mathbf{r}_{1}=\left(\begin{array}{l}5 \\ 8 \\ 2\end{array}\right)+t\left(\begin{array}{c}-1 \\ 2 \\ 3\end{array}\right), t \in \mathbb{R} $.
1. (1) Write down the coordinates of P .(a,b,c) a=   b=   c =   
(2) Find the speed of Jack's airplane in $\mathrm{ms}^{-1}$ .≈    $ms^{-1}$
2. After six seconds, Jack's airplane passes through a point Q .
(1) Find the coordinates of Q . (a,b,c) a=   b=   c =   
(2) Find the distance the airplane has travelled during the six seconds.

John's airplane passes through a point R. Its position, s seconds after it passes through $R$ , is given by $\mathbf{r}_{2}=\left(\begin{array}{l}4 \\ 4 \\ 5\end{array}\right)+s\left(\begin{array}{c}-1 \\ 5 \\ 3\end{array}\right), s \in \mathbb{R}$ . ≈    m
3. Find the coordinates where the two airplanes intersect.(a,b,c) a=   b=   c =   
4. Determine who is flying faster, Jack or John. Justify your answer.

  A particle $\mathrm{P}$ moves with velocity $\mathbf{v}=\left(\begin{array}{c}2 \\ -5 \\ 4\end{array}\right)$ in a magnetic field, $\mathbf{B}=\left(\begin{array}{l}a \\ 0 \\ 2\end{array}\right), a \in \mathbb{R}$ .
1. Given that $\mathbf{v}$ is perpendicular to $\mathbf{B}$ , find the value of a .
The force, $\mathbf{F}$ , produced by $\mathrm{P}$ moving in the magnetic field is given by the vector equation $\mathbf{F}=b \mathbf{v} \times \mathbf{B}, b \in \mathbb{R}^{+}$ .
2. Given that $|\mathbf{F}|=18$ , find the value of b .   

  Emily starts flying hjk zs*e.m6v syc0-d6e-c gxo:er drone from the roof of a residential building. The displacement of the drone atc-d ge*czy:e.-smk v6sojx60 time t seconds is given by the vector equation
$\mathbf{r}_{\mathrm{d}}=\left(\begin{array}{c}
0 \\
0 \\
42
\end{array}\right)+t\left(\begin{array}{c}
4 \\
6 \\
0.4
\end{array}\right), \quad t \geq 0 .$
where distances are measured in metres.
1. Find the position vector of Emily's drone one minute after takeoff.
Frank starts flying his quadcopter (a type of drone) from a tennis court. The displacement of the quadcopter at time s seconds is given by the vector equation
$\mathbf{r}_{\mathrm{q}}=\left(\begin{array}{c}
-420 \\
-360 \\
0
\end{array}\right)+s\left(\begin{array}{c}
10 \\
12 \\
1
\end{array}\right), \quad s \geq 0 .$
where distances are are also measured in metres.$\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)$ a =    b =    c =   
2. Find the distance between Emily and Frank. ≈    m
3. Determine if the two drone flight paths intersect, and if so, write down the point of intersection.
Emily's drone and Frank's quadcopter started flying at the same time.$\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)$ a =    b =    c =   
4. State whether the two drones actually collide. Justify your answer.
5. (1) Find the time when Emily's drone is closest to Frank's quadcopter. $t_{min}$ ≈    seconds
(2.)Calculate the minimum distance between the two drones. $d_{min}$ ≈    m

  A land surveyor is measuring the area of a triangular plothwbt 2d5xi3xxwb i/m * k6gn-7 of land on a smooth hill. She uses a theodolite to determine the coordinates of the corners of th2x-ti i3wb/kh7*wnbm6xd xg 5e land and finds them to be (-8,20,10),(16,8,7) and (-20,-8,3) , where the coordinates are in metres from a fixed origin where she stands.
Using vectors, determine the area of the land.    $m^2$

  A land surveyor is measuring the area of a triangr6 zff3 /rh5cbjs d)4wular plot of land on a smooth hill. She uses a theodolite to determine the coordinates of cw )4d/r3 bjr5fz6 sfhthe corners of the land and finds them to be (-8,20,10),(16,8,7) and (-20,-8,3) , where the coordinates are in metres from a fixed origin where she stands.
Using vectors, determine the area of the land.A lighthouse positioned at $\mathrm{O}(0,0)$ is located at the centre of a ring shaped coral reef. A sailboat nearby, initially outside the reef, travels at a constant speed of $8 \sqrt{5} \mathrm{~km} \mathrm{~h}^{-1}$ in the direction $\mathbf{d}=\mathbf{i}+2 \mathbf{j}$ . Distances are measured in kilometres.
1. Find the velocity vector of the sailboat, in terms of $ \mathbf{i}$ and $\mathbf{j}$ .
Initially, the sailboat is at the point \mathrm{A}(4,-12) .
2. Find, in terms of $\mathbf{i}$ and $ \mathbf{j}$ , its:
(1) initial position vector;
(2)position vector after t hours, $t \geq 0$ .
3. Find the time when the sailboat is closest to the lighthouse.
The ring-shaped coral reef has a radius of 10 $\mathrm{~km}$ . t =    hours
4. Determine whether the sailboat will hit the reef. Justify your answer.

  Grace and Henry are pl,3ow0s b+ tq3 q4 cilzml,xrqud,q. +baying on a youth soccer field. Grace kicks a ball from $\mathrm{A}(4,3)$ with velocity vector $-5 \mathbf{i}+10 \mathbf{j}$ . At the same time, Henry kicks a ball from $\mathrm{B}(24,13)$ at 15$ \mathrm{~m} \mathrm{~s}{ }^{-1}$ in the direction $-4 \mathbf{i}+3 \mathbf{j}$ . The times and distances are measured in seconds and metres respectively.
1. Find, in terms of $\mathbf{i}$ and $\mathbf{j}$ , the :
(1) velocity vector of Henry's ball;
(2) position vector of Henry's ball after t seconds, $ t \geq 0$ .
2. Find the distance between the two balls after 2 seconds.    m
3. (1) Calculate the minimum distance between the balls.
(2) Determine the position of Grace's ball when this occurs.

  A civil engineer is measuring the area of sloped land to be used for a res f wb b0jyoy):(w7(toridential property development project. She uses a total station theodolite device to determine the coordinates of the corners of the land and finds them to be (15,25,9),(20,5,5),(-10,-10,2) and (-15,10,6) where the coordinates are in metres fr:ywt b70r j(wyo)ob(f om a fixed origin where she is standing. Using vectors, determine the area of the land.    $m^2$

  A sailboat is traveling in a straight line tl(ftr3 zx -fm0 f--xagiven by the parametric equations x=6+4.5 t and y=9 t-11 , where t is the travel time, in hours, after 16:30. The pt zt x0f(mf- lxa--fr3ositive x -axis is due east and the positive y -axis is due north. Distances are measured in kilometres.
1. Write down the position of the sailboat at 16:30.(a,b) a=   b=  
2. (1) Write down the velocity vector of the sailboat.$\begin{pmatrix}
a \\
b
\end{pmatrix}$ a =    b =   
(2) Find the speed of the sailboat.
A lighthouse is located at $\mathrm{P}(22,0)$ .≈    $km\,h^{-1}$
3. Find the distance of the sailboat from the lighthouse at 19:30.≈    km
4. Find the time, to the nearest minute, when the sailboat is:
(1) closest to the lighthouse;
(2) directly to the north of the lighthouse.

  The Great Pyramid of Giza can be modelled by a right-pyay6q.tl y1nnnvdecyxc2 0 z )9(sw-h1 ramid with a square base. A scale model of1 c sqav0 w6).zntdce9-(yny xny 1l2h the Great Pyramid of Giza is shown below.



The vertices $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ have coordinates $\mathrm{A}$(1,5.5,0), $\mathrm{B}$(6.5,0,7) and $\mathrm{C}$(12,5.5,0) relative to a fixed origin O near the pyramid. All distances are measured in centimetres.
1. 1. Find $\overrightarrow{\mathrm{AB}}$ .
2. Find $\overrightarrow{\mathrm{AC}}$ .
2. Find $\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}$ .
3. Hence, or otherwise, find the surface area of the scale model, not including the base. ≈    $cm^2$

  The displacement, from a fixed origin at sea level, of a passenger riding otlm92fb iyr4 hssmu8 * x5zg +yvp3,x1n a gondola ski lift is modelled by the veypix9bm,l1+vg sztx82*r fsu 54h3 ymctor equation

$\mathbf{r}=\left(\begin{array}{c}
244 \\
12 \\
1650
\end{array}\right)+t\left(\begin{array}{l}
2.4 \\
5.2 \\
1.8
\end{array}\right)$
where t is the travel time, in seconds, starting at the base station and ending at the top station. Distances are measured in metres. Bill boards the ski lift from the base station and travels to the top station.
1. Write down Bill's :
(1) initial position;(a,b,c) a=   b=   c =   
(2) velocity vector.$\left(\begin{array}{l}
a \\
b \\
c
\end{array}\right)$ a=   b=   c =   
2. Find the speed, in metres per second, of the gondola ski lift.
It takes k minutes for Bill to reach the top station, which has a vertical height of 2352 metres.    $ms^{-1}$
3. Find the value of k .   
4. Find:
(1) Bill's terminal position;(a,b,c) a=   b=   c =   
(2) the length of Bill's ride.≈    m
5. Find Bill's position when he is closest to a skier resting on a hill at the point $\mathrm{P}$(610,962,2020) .(a,b,c) a=   b=   c =   

  WaveRunner X leaves fro:icm,/s c : 2gcq0dytqm $\mathrm{A}$(-265,141) and travels with velocity vector 10 $\mathbf{i}-4 \mathbf{j}$ . At the same time, WaveRunner $\mathrm{Y} $ leaves from $\mathrm{B}$(-180,-103) and travels at 19.5 $\mathrm{~m} \mathrm{~s}{ }^{-1} $ in the direction $5 \mathbf{i}+12 \mathbf{j} $. Times and distances are measured in seconds and metres respectively.
1. Determine if the two WaveRunners are travelling perpendicular to each other.
2. Find, in terms of $\mathbf{i}$ and $\mathbf{j}$ , the :
(1) velocity vector of WaveRunner Y . $\mathbf{v}_{\mathrm{Y}}=a \mathbf{i}+b \mathbf{j} $ a =    b =   
(2) position vector of WaveRunner $\mathrm{Y}$ after t seconds, $t \geq$ 0 .
3. Find the point at which the paths of the WaveRunners intersect.(a,b) a=   b=  
4. Calculate the distance between WaveRunners X and Y when WaveRunner X passes through the intersection point.    m
5. Find the time, in seconds, when WaveRunner X is closest to WaveRunner Y's starting point. ≈    seconds

  Nikola is testing his new autonomous underwater t:uex (xzgi)w;f, rx/ vehicle (AUV) for diving and surfacing (t e/rxx ;i)uxgz w:,fin the Black Sea.
After submersion in the sea, the AUV begins traveling with a speed of 1.5 $\mathrm{~m} \mathrm{~s}^{-1}$ in the direction $2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$ , where the unit vector \mathbf{i} is due east, $\mathbf{j}$ is due north and $\mathbf{k}$ is perpendicular to the sea surface. The time is measured in seconds and distances are measured in metres.
1. (1) Find the velocity vector of the AUV.$\mathbf{v}_{\mathrm{d}}=a\mathbf{i}+b \mathbf{j}+c\mathbf{k}$ a =    b =    c =   
(2) Write down the diving speed of the AUV.
Nikola's yacht is at $\mathrm{O}$(0,0,0) . The AUV passes under the yacht at time t=0 and its position vector at this moment is $-30 \mathbf{k}$ .    $ms^{-1}$
2. Find the displacement of the AUV relative to the yacht one minute later.

At time t=60 , the AUV turns $90^{\circ}$ in the clockwise direction, maintaining the same horizontal speed, and begins surfacing at $0.75 \mathrm{~m} \mathrm{~s}^{-1}$ .$\mathbf{r}_{\mathrm{d}}=a\mathbf{i}+b \mathbf{j}+c\mathbf{k}$ a =    b =    c =   
3. 1. Find the velocity vector of the AUV at the moment it begins surfacing.$\mathbf{v}_{\mathrm{s}}=a\mathbf{i}+b \mathbf{j}+c\mathbf{k}$ a =    b =    c =   
(2) Hence, write down the displacement of the AUV relative to the yacht at time $t^{\prime} $ seconds, where $t^{\prime}=0 $ is the time at which the AUV begins surfacing.
4. Find the time it takes for the AUV to reach the sea surface from the moment it begins surfacing.    seconds
5. Find the distance of the AUV from the yacht when it reaches the sea surface.    m

  Jack is a student pilot flying a Cessna 172 qhajx(deq 22;Skyhawk and his flight is being monitored by the air traffic co;hq(2 j a2qxedntrol centre. The aircraft's position is given by the coordinates ( x, y, z) , where x and y are the aircraft's displacement east and north of the airport, and z is the height of the aircraft above the ground. All displacements are given in kilometres.
The aircraft's velocity is given by the vector $\left(\begin{array}{c}-160 \\ -64 \\ -12\end{array}\right) \mathrm{km} \mathrm{h}^{-1}$ .
At 11 : 00 the air traffic control centre detects Jack's aircraft at a position 40 $\mathrm{~km}$ east and 16 $\mathrm{~km} $ north of the airport, and at a height of 4.5 $\mathrm{~km}$ . Let t be the length of time, in hours, from 11: 00 .
1. Write down a vector equation for the aircraft's displacement, $\mathbf{r}$ , in terms of t .
2. Given that Jack's aircraft continues to fly at the same velocity,
(1) verify that it will pass directly over the airport;$\mathbf{r}=\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)+t\left(\begin{array}{c}
d \\
e \\
f
\end{array}\right)$a =    b =    c =    d =    e =    f =   
(2) find the time at which this happens;
(3) calculate its height at this point.    km

When Jack's aircraft is 3.3 $\mathrm{~km} $ above the ground, it continues to fly on the same bearing but adjusts the angle of descent so that it will land at the point $\mathrm{P}$(0,0,0) .
3. (1) Find the time at which the aircraft is 3.3 $\mathrm{~km}$ above the ground.
(2) Find the direct distance of the aircraft from the airport at this point. the vector $\left(\begin{array}{c}-160 \\ -64 \\ p\end{array}\right) \mathrm{km} \mathrm{h}^{-1}$ . ≈    km
4. Find the value of p .   

  The US Defense Force is testing a new drone. The drone is z1q9oayw0n )gcontrolled by a remote ground control system. The drone's position is given by the coordinates ( x, y, z) , where x and y are the drone's displacement east and north of an airbase, and z is the height of the drone above the ground. All displacements are given in9 qzg1nw)oa 0y kilometres.
The drone's velocity is given by the vector $\left(\begin{array}{c}-80 \\ -240 \\ -15\end{array}\right) \mathrm{km} \mathrm{h}^{-1}$ .
At 15: 00 the remote ground control system detects the drone at a position 32 \mathrm{~km} east and 96 $\mathrm{~km}$ north of the airbase, and at a height of $8 \mathrm{~km}$ . Let t be the length of time, in hours, from 15: 00 .
1. Write down a vector equation for the drone's displacement, $\mathbf{r}$ , in terms of t .$\mathbf{r}=\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)+t\left(\begin{array}{c}
d \\
e \\
f
\end{array}\right)$a =    b =    c =    d =    e =    f =   
2. Given that the drone continues to fly at the same velocity,
(1) verify that it will pass directly over the airbase;
(2) find the time at which this happens;
(3) calculate its height at this point.
The drone continues to fly at the same velocity and descends to a height of 5 $\mathrm{~km}$ .    km
3. Find the time at which this happens.
4. Calculate the direct distance of the drone from the airbase at this point.
After descending to a height of 5 $\mathrm{~km}$ , the drone continues to fly on the same bearing but adjusts the angle of descent so that it will land at the point Q(0,0,0) .
The drone's velocity, after the adjustment of the angle of descent, is given by
the vector $\left(\begin{array}{c}-80 \\ -240 \\ q\end{array}\right) \mathrm{km} \mathrm{h}^{-1} $. ≈    km
5. Find the value of q .   

  Aircraft $\mathrm{X}$ is flying south-east on a bearing of $135^{\circ}$ at a speed of 920 $\mathrm{~km}$ $\mathrm{~h}^{-1}$ and is ascending at a rate of 1.2 $\mathrm{~km} \mathrm{~h}^{-1}$ . At 16: 00 the aircraft is directly above a tracking station at an altitude of 8.8 $\mathrm{~km}$ .
1. Find an expression for the displacement of Aircraft X from the tracking station at time t hours after 16: 00 .

Aircraft $\mathrm{Y}$ is flying on a bearing of $120{ }^{\circ}$ at a speed of 840 $\mathrm{~km}$ $\mathrm{~h}^{-1}$ and is descending at a rate of 2.4 $\mathrm{~km} \mathrm{~h}^{-1}$ . At 16: 45 the aircraft is directly above a tracking station at an altitude of 12.4 $\mathrm{~km}$ .$\begin{aligned}
\mathbf{r}_{X} &
& =\left(\begin{array}{c}
a \\
b \\
c
\end{array}\right)+t\left(\begin{array}{c}
d \sqrt{e} \\
f \sqrt{g} \\
h
\end{array}\right)
\end{aligned} $ a =    b =    c =    d =    e =    f =    g =    h =   
2. Find an expression for the displacement of Aircraft Y from the tracking station at time t hours after 16: 00 .

At time k hours after 16: 00 the two aircrafts reach the same cruising altitude.
3. Find the value of k .   

At t=k , Aircraft $\mathrm{X}$ is at point $\mathrm{A}$ and Aircraft $\mathrm{Y}$ is at point $\mathrm{B}$ .
4. Find the coordinates of :
1. A;$(a \sqrt{b},c \sqrt{d}, e)$ a =    b =    c =    d =    e =   
2. B. $(a \sqrt{b},c,d)$ a =    b =    c =    d =   
5. Hence, find the distance between the two aircrafts when they reach the same cruising altitude. ≈    km

  Adriano is riding a ,o;kdrs ;uiu ( e,1cjrq:rwe (skateboard in a parking lot. His position vector from a fixed originesk: 1cu;;,orwjru(( eqdi, r $\mathrm{O}$ at time t seconds is modelled by
$\left(\begin{array}{l}
x \\
y
\end{array}\right)=\left(\begin{array}{l}
a \ln (t+b) \cos t \\
a \ln (t+b) \sin t
\end{array}\right)$
where a and b are non-zero constants to be determined. All distances are in metres.
1. Find the velocity vector at time t .
2. Given that a>0 , show that the magnitude of the velocity vector at time t is given by a $\sqrt{\frac{1}{(t+b)^{2}}+(\ln (t+b))^{2}}$ .
At time t=0 , the velocity vector is $\left(\begin{array}{c}2 \\ 2.773\end{array}\right) $.
3. Find the value of a and the value of b . a =    b =   
4. Find the magnitude of the velocity vector when t=3 .
At point $\mathrm{P}$ , Adriano is riding parallel to the y -axis for the first time. ≈    $ms^{-1}$
5. Find $|\mathrm{OP}| $. ≈    m

  Mia is riding a bicycle in a town square. Her position vector from a fixedrn-(5kem i/z twze2i r:dr .m7 origin rr:/rtew.dk miz2 7nemzi( 5-\mathrm{O} at time t seconds is modelled by

$\left(\begin{array}{l}
x \\
y
\end{array}\right)=\left(\begin{array}{l}
r e^{k t} \sin t \\
r e^{k t} \cos t
\end{array}\right)$

where r and k are non-zero constants to be determined. All distances are in metres.
1. Find the velocity vector at time t .
2. Given that r>0 , show that the magnitude of the velocity vector at time t is given by $r e^{k t} \sqrt{k^{2}+1}$ .

At time t=0 , the velocity vector is $\left(\begin{array}{c}6 \\ -2.4\end{array}\right)$ .
3. Find the value of r and the value of k . r =    k =   
4. Find Mia's speed after 4 seconds.

At point $\mathrm{P}$ , Mia is riding parallel to the x -axis for the first time.≈    $ms^{-1}$
5. Calculate the distance from Mia's starting position to point P .≈    m