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.

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

Sample Solution