First slide

Vectors

Question

A sail boat sails 2 km due East, 5 km $37^{0}$ 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.

Moderate

A

3 km, North
B

4 km, South
C

5 km, East
D

3 km, West

Solution

(a) Given resultant displacement is 6 km due east
Which may be expressed as $6 \hat{i}$
Given displacement are
(i) 2 km due east or $2 \hat{i}$
(ii) 5 km $37^{0} south of east its components$
along x-axis = 5 cos $37^{0} = 5 \times \frac{4}{5} = 4 \hat{i}$
along y-axis = 5 sin $37^{0} = 5 \times \frac{3}{5} = - 3 \hat{j}$
(iii) unknown displacement let it be $a \hat{i} + b \hat{j}$
$∴ 2 \hat{i} + 4 \hat{i} - 3 \hat{j} + a \hat{i} + b \hat{j} = 6 \hat{i}$
$∴ 6 \hat{i} + a \hat{i} - 3 \hat{j} + b \hat{j} = 6 \hat{i}$
comparing the two sides
a = 0 and b = 3
$∴ unknown displacement = 3 \hat{j}$

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