Modelling with Vectors (DP IB Maths: AI HL)

Two spaceships , in a 3D virtual reality game, are moving such that their positions relative to a fixed point  at time seconds, , are defined by the position vectors  and  respectively.

Show that the two spaceships are on course to collide at point  and write down the coordinates of .

Spaceship  reduces its velocity such that its position vector is now given by

Show that spaceship is still travelling in its original direction.

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

Hence find the distance between the two spaceships when spaceship  is at .

A car is moving with constant velocity along the line with equation A bird is perched at the point  and at , starts to fly at a constant velocity in the direction of the vector .

All distances are measured in metres and time in seconds. The base vectors  and  represent due east and due west respectively and the base vector  points upwards.

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, , measured in kilometres per minute, where

 . 

A fixed public telescope is located at a point  relative to an origin O.  At time  minutes the hawk is at the point and at time  minutes the hawk is vertically above the telescope.

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

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

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

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

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

The  and  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 seconds and reach the ground at the same place as the mouse.

During their subsequent motion, find the speed of

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 , through a second point  and continues walking in a straight line. At the same time a leaf located at  falls from its starting position in a tree and moves with constant velocity in the direction of the vector .   

The  and  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 . At the time, t seconds, the tractor is moving relative to an origin, O. At  the position vector of the tractor is . 

Find

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 . The ball moves freely under gravity. The velocity of the subsequent motion of the ball can be modelled by the vector

where is the horizontal component of the initial velocity and 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 .

Find

(i)

the displacement vector of the ball at the time, ,

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  seconds, is given by the vector

Find

A turtle is riding on the back of the whale and at the time seconds begins to swim away from it with velocity vector  , where  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

  and 

at time  minutes after take-off, . All distances are in metres.

A third drone Z begins its flight at  and its position relative to the observation centre is given by  

Each drone can observe a circular area of ground,   such that  where  is the height of the drone above the ground in metres.

A car is moving at a constant speed of 15 ms^-1 in the direction parallel to the vector  Two birds are perched at points  and . 

At , the car is located at  and the bird at point A starts to fly at a constant velocity of   ms^-1. The bird at point B begins to fly at a constant velocity in the direction of the vector  when . 

When bird A reaches the position of , both birds and the car lie in a straight line.

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

Points and 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  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 coordinates are all positive. 

• The distance from point C  to point S is exactly  parsecs. 

• Points  and   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

Taking the origin to be the point at which the oyster is sitting, the unit vectors and  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 is double that of when it is at its maximum height, find

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

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

The boat leaves its mooring point at time  seconds and 5 minutes later is at the point with coordinates . 

Find

(i)

the values of  and , 

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

The component represents horizontal motion and the  component represents vertical motion. The start of the runway is considered the origin and the runway runs along the
­horizontal axis. When seconds the velocity of the plane is  and the plane is 27 metres vertically above the start of the runway.  

Find

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, moves with velocity at time t seconds given by the equation

The second worm,  has position vector given by

 . 

All distances are in metres and time is in seconds.

Find the velocity vector of 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 , find

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

where and .

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

Given that at time seconds the spider is moving parallel to the y-axis with a speed of   , find the values of  and .

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