Modelling with Vectors (DP IB Applications & Interpretation (AI): HL): Exam Questions

Two spaceships $\mathrm{A}\mathrm{and}\mathrm{B}$, in a 3D virtual reality game, are moving such that their positions relative to a fixed point $\mathrm{O}$ at time $t$ seconds, $0\le t<30$, are defined by the position vectors $r_A=\left(\begin{array}{c}2\\ -3.5\\ 1\end{array}\right)+t\left(\begin{array}{c}1.2\\ 0.5\\ 2\end{array}\right)$ and $r_B=\left(\begin{array}{c}-2\\ 4\\ 9.5\end{array}\right)+t\left(\begin{array}{c}2\\ -1\\ 0.3\end{array}\right)$ respectively.

Show that the two spaceships are on course to collide at point $\mathrm{P}$ and write down the coordinates of $\mathrm{P}$.

Spaceship $\mathrm{B}$ reduces its velocity such that its position vector is now given by

$r_B=\left(\begin{array}{c}-2\\ 4\\ 9.5\end{array}\right)+t\left(\begin{array}{c}1.6\\ -0.8\\ 0.24\end{array}\right)$ 

Show that spaceship $B$ is still travelling in its original direction.

Show that the distance between the two spaceships can be written as

$\sqrt{4.9476t^2-56.62t+144.5}$

Hence find the distance between the two spaceships when spaceship $\mathrm{A}$ is at $\mathrm{P}$.

A car is moving with constant velocity along the line with equation $r_c=\left(\begin{array}{c}2\\ 3\end{array}\right)+t\left(\begin{array}{c}5\\ 12\end{array}\right).$ A bird is perched at the point $\left(25,32,8\right)$ and at $t=0$ , starts to fly at a constant velocity in the direction of the vector $\left(2\mathrm{i}+31\mathrm{j}-4k\right)$.

All distances are measured in metres and time in seconds. The base vectors $\mathrm{i}$ and $\mathrm{j}$ represent due east and due west respectively and the base vector $\mathit{k}$ points upwards.

Verify that the bird does not collide with the car.

Show that at some point in time the bird will be directly above the car and state the time at which this occurs.

Hence find the distance between the bird and the car at that time.

A hawk is flying with constant velocity, $v$, measured in kilometres per minute, where

 $v=\left(\begin{array}{c}-1\\ a\\ 0.2\end{array}\right)$. 

A fixed public telescope is located at a point $A\left(1,-2\right)$ relative to an origin O.  At time $t=0$ minutes the hawk is at the point $\left(4,-1,1\right)$ and at time $t=T$ minutes the hawk is vertically above the telescope.

The $x$direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards. All distances are measured in kilometres. 

Find the value of $a$ and the height of the hawk when it is directly above the telescope.

Given that the hawk continues flying in the same direction, find 

i) the speed at which the hawk is flying,

ii) the time at which the hawk is exactly 4 km from the telescope.

An eagle is tracking a mouse on the ground that is moving with constant velocity such that its position vector relative to an origin O, at time t  seconds can be modelled by

 $\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}30\\ -10\end{array}\right)+t\left(\begin{array}{c}-2\\ 3\end{array}\right)$ 

At time $t=3$ seconds the eagle is positioned at a point exactly 28 metres vertically above the mouse. 

The $x,y$ and $z$ directions are due east, due north and vertically upwards respectively with all distances in metres.

Write down a vector equation of a line the eagle should fly in if it is to take off at the time $t=3$ seconds and reach the ground at the same place as the mouse.

During their subsequent motion, find the speed of

i) the eagle,

ii) the mouse.

Luckily for the mouse, at the point where the eagle is exactly 5 metres from it, it finds a hole to hide in.

Find the amount of time the mouse had left before the eagle would have reached it.

A person is walking in a straight line and at a constant speed. The person moves from their initial point $\left(5,1\right)$, through a second point $\left(2,-6\right)$ and continues walking in a straight line. At the same time a leaf located at $\left(1,-5,14\right)$ falls from its starting position in a tree and moves with constant velocity in the direction of the vector $a\mathrm{i}+\left(a-2\right)\mathrm{j}-4.92\mathrm{k}$.   

The $x,y$ and $z$ directions are due east, due north and vertically upwards respectively with all distances in metres. 

Given that the person is 1.7 metres tall and that they are walking on a horizontal plane, find the coordinates of the point at which the leaf lands on the top of their head.

The leaf takes 14.2 seconds to fall from its starting point in the tree to the point where it lands on the person’s head. Find the speed with which it falls.

The velocity of a tractor is given by $v=\left(\begin{array}{c}10\sin t\\ -10\cos t\end{array}\right)$. At the time, t seconds, the tractor is moving relative to an origin, O. At $t=0$ the position vector of the tractor is $\left(\begin{array}{c}-10\\ 0\end{array}\right)$. 

Find

i) the initial speed of the tractor

ii) a position vector of the tractor at time, t , relative to the origin.

Find the distance of the tractor from the origin at time, t.

Describe the shape of the path that the tractor is making.

A ball is thrown from a height of 1.2 metres with initial velocity $u=\left(\begin{array}{c}4\\ 15\end{array}\right){\mathrm{ms}}^{-1}$. The ball moves freely under gravity. The velocity of the subsequent motion of the ball can be modelled by the vector

 $v\left(t\right)=\left(\begin{array}{c}{u}_x\\ u_y-gt\end{array}\right)$ 

where $u_x$is the horizontal component of the initial velocity and $u_y$is the vertical component of the initial velocity. 

The ground below the point from which the ball was thrown can be considered the origin. It is assumed that any effects of air resistance will be negligible, the ball is modelled as a particle and $g=9.81{\mathrm{ms}}^{-2}$.

Find

(i) the displacement vector of the ball at the time, $t$ ,

(ii) the maximum height of the ball,

(iii) the angle of elevation of the ball as it was thrown.

Assuming that the ground is horizontal, find how far from the origin the ball is when it lands.

A whale is moving through the sea such that its position, at time $t$ seconds, is given by the vector

 $r=\left(\begin{array}{c}\frac{1}{3}{t}^2-\frac{1}{2}t\\ \frac{1}{2}{t}^2+4t\end{array}\right)$ 

Find

(i) the initial speed of the whale,

(ii) the speed of the whale after 10 seconds,

(iii) the acceleration vector of the whale

A turtle is riding on the back of the whale and at the time $t=10$seconds begins to swim away from it with velocity vector $v=\frac{1}{2}{T}^2\mathrm{i}+\frac{3}{4}{T}^2\mathrm{j}$  , where $T=t-10$ seconds. The whale continues to swim along the same path. 

Find the distance between the whale and the turtle 10 seconds after they separated.

Two drones X and Y are being flown over an area of rainforest to look for signs of illegal logging. Their positions relative to the observation centre, are given by

$r_x=\left(\begin{array}{c}-3\\ 1.6\\ 2.5\end{array}\right)+t\left(\begin{array}{c}2\\ -2\\ 1\end{array}\right)$  and $r_y=\left(\begin{array}{c}2.5\\ 0\\ -2\end{array}\right)+t\left(\begin{array}{c}1.5\\ 6\\ 4\end{array}\right)$

at time $t$  minutes after take-off, $0\le t<20$. All distances are in metres.

Verify that the two drones will not collide.

Find the shortest distance between the two drones and the time at which it occurs.

A third drone Z begins its flight at $t=8$ and its position relative to the observation centre is given by $r_z=\left(\begin{array}{c}2\\ 1.5\\ 4.5\end{array}\right)+t\left(\begin{array}{c}3\\ 4\\ 1\end{array}\right)$ 

Each drone can observe a circular area of ground,  $A,$ such that $A=1.8h^2$ where $h$ is the height of the drone above the ground in metres.

Show that the area of ground that can be observed by drone Z five minutes after it takes off overlaps with the area of ground that can be observed by drone Y at that time.

A car is moving at a constant speed of 15 ms^-1 in the direction parallel to the vector $3\mathrm{i}-6\mathrm{j}.$  Two birds are perched at points $\mathrm{A}(17,28,16)$  and $\mathrm{B}\left(-48,128,26\right)$. 

At $t=0$, the car is located at $\left(2,4,0\right)$  and the bird at point A starts to fly at a constant velocity of  $\frac{7\sqrt{365}}{10}$ ms^-1. The bird at point B begins to fly at a constant velocity in the direction of the vector $52\mathrm{i}-60\mathrm{j}-9k$ when $t=1.2$. 

When bird A reaches the position of $\left(44,-24,4\right)$, both birds and the car lie in a straight line.

Find the equation of the line along which the birds and car lie.

Find the speed at which bird B is travelling.

Consider the following diagram depicting imaginary lines connecting five points in space:

Points $A,B,C$ and $D$ are the locations, respectively, of the stars Arccirclus, Betacarotjuse, α-Capella and Denomineb.  Point S is the location of the Stellamortis battle station, a planet-killing atrocity being built by the evil Galactic Imperium.  Coordinates are given relative to an origin point in accordance with the standard $x,y,z$ coordinate system, and the units for all coordinates are parsecs. 

The forces of the Star Rebellion are prepared to launch a strike to destroy the battle station, but they are unsure of its exact location.  According to data recovered from a smuggled droid, however, the following facts are known about the location of point S : 

• Point S is in the First Octant of the galaxy, where $x,y\mathrm{and}z$ coordinates are all positive. 

• The distance from point C  to point S is exactly  $45\sqrt{2}$parsecs. 

• Points $B,C,D$ and $S$  form the base of a pyramid, with its apex at point A.

• The point on BD closest to point A is also the point where the two diagonals of the pyramid’s base intersect.

As the rebellion’s Chief Mathematician, it is your job to use the information provided to find the exact coordinates of point S.  The fate of the galaxy is in your mathematical hands!

An oyster on the edge of a coral reef projects a microbubble into a jet stream and its subsequent motion can be modelled as a position vector. The microbubble reaches a maximum height and then moves back downwards in front of the oyster and continues down into the sea below.

The acceleration of the microbubble can be modelled by the vector

 $a=\left(0.4\mathrm{i}-0.6t\mathrm{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$ 

Taking the origin to be the point at which the oyster is sitting, the unit vectors $\mathrm{i}$ and $\mathrm{j}$ are a displacement of 1 m along the horizontal and vertical axis of a Cartesian coordinate system respectively.

Given that it takes 5 seconds until the microbubble is at the same horizontal height as the oyster again, and that the horizontal distance of the microbubble from the oyster at $t=5$ is double that of when it is at its maximum height, find

(i) the maximum height above the oyster that the microbubble reaches,

(ii) the position vector of the microbubble at time, t .

The seabed is 22 metres below the level of the oyster. Find the speed of the microbubble at the moment when it hits the seabed.

A boat is moving such that its position vector when viewed from above at time t  seconds can be modelled by

 $r=\left(\begin{array}{c}10-a\sin \left(\frac{\mathrm{\pi t}}{600}\right)\\ b\left(1-\cos \left(\frac{\mathrm{\pi t}}{600}\right)\right)\end{array}\right)$ 

with respect to a rectangular coordinate system from a point O, where the non-zero constants $a$  and $b$ can be determined. All distances are given in metres. 

The boat leaves its mooring point at time $t=0$ seconds and 5 minutes later is at the point with coordinates $\left(-20,40\right)$. 

Find

(i) the values of $a$ and $b$, 

(ii) the displacement of the boat from its mooring point.

Find the velocity vector of the boat at time t seconds.

Find the time that the boat returns to its mooring point and the acceleration of the boat at this moment.

A small stunt plane is heading in to land at an airport with acceleration given by the vector

 $a=\left(\begin{array}{c}-0.06\\ -0.03t\end{array}\right){\mathrm{ms}}^{-2}$ 

The $\mathrm{i}$ component represents horizontal motion and the $\mathrm{j}$ component represents vertical motion. The start of the runway is considered the origin and the runway runs along the ­horizontal axis. When $t=10$ seconds the velocity of the plane is $0.9\mathrm{i}-3.5\mathrm{j}{\mathrm{ms}}^{-1}$ and the plane is 27 metres vertically above the start of the runway.  

Find

(i) the time in seconds at which the stunt plane lands on the runway,

(ii) the distance of the stunt plane from the start of the runway when it lands.

Find the speed with which the stunt plane lands.

At the moment of landing, this particular type of stunt plane needs to have a deceleration of between 0.4 and 0.5 ms^-2.

Decide whether the stunt plane has landed within the safe landing limits.

Two children are observing the movement of some worms in their garden. The worms are placed on the ground at the same time and begin to move instantly. The first worm, $W_1$ moves with velocity at time t seconds given by the equation

$v=\left(\begin{array}{c}{e}^{-0.4t}\left(a\cos t-b\sin t\right)\\ e^{-0.4t}\left(a\sin t+b\cos t\right)\end{array}\right).$ 

The second worm, $W_2$ has position vector given by

 $r=\left(\begin{array}{c}4e^{-0.5t}\cos t\\ 3e^{-0.2t}\sin t\end{array}\right)$. 

All distances are in metres and time is in seconds.

Find the velocity vector of $W_2$at time t seconds.

Given that both worms are travelling parallel to each other in the same direction and at the same speed at time $t=20$, find

(i) the values of a and b,

(ii) the speed at which the two worms are travelling at this moment.

A spider starts from the origin and begins to weave a web such that her velocity vector at time t  seconds with respect to a rectangular coordinate system can be modelled by

 $v=\left(\begin{array}{c}a\sin \left(bt\right)+\sin \left(t\right)\\ \cos \left(t\right)-a\cos \left(bt\right)\end{array}\right)$ 

where $a<0$ and $0<b<0.15$.

Find an expression for the position vector of the spider at time t, in terms of $a$ and $b$.

Given that at time $t=9\mathrm{\pi }$ seconds the spider is moving parallel to the y-axis with a speed of  $\frac{7}{3}{\mathrm{ms}}^{-1}$ , find the values of $a$ and $b$.

Find the earliest time at which the spider is weaving its web parallel to the x-axis.