Question

Find the north and east components of the displacement for the hikers shown in Figure.

Short answer

The east component is 3.8 km

The north component is 3.2 km

Step-by-step solution

Vector definition

A vector is a quantity that has both magnitude and direction. Displacement, velocity, force, torque, and other vector quantities are some examples of vector quantities.

Determine the displacement of the hiker towards the north component

In triangle ABC:

$\begin{array}{rcl}\angle ABC& =& {90}^{°}-{40}^{°}\\ & =& {50}^{°}\end{array}$

$\mathrm{BC}=5\mathrm{km}$

AC = Base

BC = Hypotenuse

AB = Perpendicular

For the north component, in the right-angled triangle:

$\begin{array}{rcl}\cos \theta & =& \frac{base}{hypotenous}\\ \cos {50}^{°}& =& \frac{base}{5}\\ 0.64& =& \frac{base}{5}\\ base& =& 3.2km\end{array}$

Therefore, towards the north component, the displacement of the hiker is 3.2 km.

Step 3: Determine the displacement of the hiker towards the east component

For the east component, in a right-angled triangle:

$\begin{array}{rcl}\sin \theta & =& \frac{Perpendicular}{hypotenous}\\ \sin {50}^{°}& =& \frac{perpendicular}{5}\\ 0.77& =& \frac{perpendicular}{5}\\ Perpendicular& =& 0.77\times 5\\ perpendicular& =& 3.8km\end{array}$

Therefore, towards the east component, the displacement of the hiker is 3.8 km.