Question worth 17 marks

An anti-whaling boat, the BlueYonder, is travelling in perfectly calm seas at
a speed (relative to the surrounding sea) of 40 km h−1 on a bearing of 160◦. It
is aiming to cross the path of a whaling ship, the Narwhal, which is sailing in
perfectly calm seas at a speed (relative to the surrounding sea) of 30 km h−1
on a bearing of 70◦(so that their paths should cross at right angles).
Take unit vectors i to point east and j to point north.

(a) Express the velocity b of the BlueYonder and the velocity n of the
Narwhal in component form, both relative to the sea, giving the
numerical values in km h−1 to one decimal place. [6]

(b) Unknown to both vessels, a strong current arises and this has the effect
of applying an additional velocity c to both vessels, of 15 km h−1 in the
direction from west to east. Write the velocity c in component form,
and hence express the resultant velocities of the BlueYonder and the
Narwhal in component form, giving numerical values to one decimal
place. [3]

(c) Find the magnitude of the resultant velocity of each vessel. Give
magnitudes in km h−1 to one decimal place. [4]

(d) Hence find the actual angle between the courses of the two vessels,
taking into account the effect of the current. Give your answer to the
nearest degree.
(Hint: use the dot product of the resultant velocity vectors.) [4]