Monitoring Currents

In this question, all distances are measured in kilometres.

2 deep sea ocean current monitors, A and B, have position vectors (-1, 7, k) and (4, 1, 10) respectively, relative to a fixed origin. Given that the distance from A to B is \(5\sqrt{5}\)km,

a) Find the possible values of the constant k.

[3 marks]

b) For the larger value of k, find the unit vector in the direction of \(\overrightarrow{\text{OA}}\)

[3 marks]

Colliding Currents

2 deep sea ocean currents meet.

By modelling one current as the positive y axis.

a) Find the angle that the second current, with vector \(4\mathbf{i – j}\), makes with the first current when they meet.

[3 marks]

A Speedy Sea Current

An ocean current separates into 2 different currents at a small island that can be modelled as the origin. Current A heads due south and current B heads on a bearing of 100˚. 

An oceanographer wants to measure the relative surface speeds of these two separate currents, by placing buoys in them off the island and measuring the distance travelled by the buoy in 20 hours.

After 20 hours, the buoy in current A has travelled 150km whilst the buoy in current B has travelled 250km.

 Find the average speeds of the two ocean currents.

[2 marks]

b) Calculate the final distance between the buoys.

[2 marks]

c) Θ is the final bearing of the buoy in current B from the buoy in current A. Show that \(\frac{\sin\theta}{\sin{80}} = 0.93\) to 2 decimal places.

[3 marks]

d) Find the area of the triangle with the island and the two buoys as its

[3 marks]