Question

A bird moves with velocity 20${ms}^{-1}$ in a direction making an angle of ${60}^0$ with the eastern line and ${60}^0$ with vertical upward. Represent the velocity vector in rectangular form.

Answer 1

Given,

$velocity=20m/s$

Eastern angle $={60}^0$

Vertical angle $={60}^0$

So Let $\alpha$, $\beta$ and $r$ be the angle substended by the $\overrightarrow{V}$

So,

$cos\alpha ={60}^0=\frac{1}{2}$

$cos\beta ={60}^0=\frac{1}{2}$

Since we know that,

$co{s}^2\alpha +co{s}^2\beta +co{s}^{2}r=1$

$(\frac{1}{2}{)}^2+\frac{1}{2}{)}^2+co{s}^{2}r=1$

$\frac{1}{4}+\frac{1}{4}+co{s}^{2}r=1$

$co{s}^{2}r=1-\frac{1}{2}$

so, $cosr=\pm \frac{1}{\sqrt{2}}$

Thus, the $\overrightarrow{V}=Vcos\alpha \hat{i}+Vcos\beta \hat{j}+Vcosr\hat{k}$

$20\times \frac{1}{2}\hat{i}+20\times \frac{1}{2}\hat{j}+20\times \frac{1}{\sqrt{2}}\hat{k}$

$=10\hat{i}+10\hat{j}+10\sqrt{2}\hat{k}$

Thus,

$\overrightarrow{V}=10\hat{i}+10\hat{j}+10\sqrt{2}\hat{k}$