3.9 Modelling with Vectors

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 . 

(i)

the values of  and , 

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.  

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.

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.

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 .