Question

A vector $\overrightarrow{\mathrm{A}}$ makes an angle of 20° and $\overrightarrow{\mathrm{B}}$ makes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find the resultant.

Answer 1

From the above figure, we have:
Angle between $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ = 110° − 20° = 90°
$\left|\overrightarrow{\mathrm{A}}\right|=3\mathrm{m}\mathrm{and}\left|\overrightarrow{\mathrm{B}}\right|=4\mathrm{m}$
Magnitude of the resultant vector is given by
$$\begin{gathered} \mathrm{R}=\sqrt{{\mathrm{A}}^2+{\mathrm{B}}^2+2\mathrm{AB}\cos \mathrm{\theta }} \\ =\sqrt{3^2+4^2+2\times 3\times 4\times \cos 90°} \\ =5\mathrm{m} \end{gathered}$$

Let β be the angle between $\overrightarrow{\mathrm{R}}\mathrm{and}\overrightarrow{\mathrm{A}}$.
$$\begin{gathered} \mathrm{\beta }={\tan }^{-1}\left(\frac{A\sin 90°}{A+B\cos 90°}\right) \\ ={\tan }^{-1}\left(\frac{4\sin 90°}{3+4\cos 90°}\right) \\ ={\tan }^{-1}\frac{4}{3} \\ ={\tan }^{-1}\left(1.333\right) \\ =53° \end{gathered}$$

Now, angle made by the resultant vector with the X-axis = 53° + 20° = 73°
∴ The resultant $\overrightarrow{\mathrm{R}}$ is 5 m and it makes an angle of 73° with the x-axis.