A sail boat sails 2 km due East, 5 km 37^o South of East and finally an unknown displacement. If the final displacement of the boat from the starting point is 6 km due East, determine the third displacement.

Given resultant displacement is 6 km due east
Which may be expressed as $6\widehat{i}.$

Given displacements are

(i) 2 km due east or $2\widehat{i}$
(ii) 5 km 37^o south of east its components
$\begin{array}{l}\text{ along }x\text{-axis }=5\cos {37}^{\circ }=5\times \frac{4}{5}=4\widehat{i}\\ \text{ along }y\text{-axis }=5\sin {37}^{\circ }=5\times \frac{3}{5}=-3\widehat{j}\end{array}$
(iii) unknown displacement let it be $a\widehat{i}+b\widehat{j}$
$\begin{array}{l}\therefore 2\widehat{i}+4\widehat{i}-3\widehat{j}+a\widehat{i}+b\widehat{j}=6\widehat{i}\\ \therefore 6\widehat{i}+a\widehat{i}-3\widehat{j}+b\widehat{j}=6\widehat{i}\end{array}$
comparing the two sides
a=0 and b=3

$\therefore \text{ Unknown displacement }=3\widehat{j}$