Question

If a vector $\overrightarrow{A}$ having magnitude $15\text{unit}$ makes an angle ${30}^{\circ }$ North of East and another vector $\overrightarrow{B}$ having magnitide $10\text{unit}$ makes an angle ${45}^{\circ }$ North of west. Find the magnitude and direction of the resultant.
(Given, $\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}$ where $\overrightarrow{R}$ is resultant vector)
[Take $\cos {30}^{\circ }=0.866,\sin {30}^{\circ }=0.5,$$\cos {45}^{\circ }=\sin {45}^{\circ }=0.707$ and ${\tan }^{-1}\left(2.46\right)={67.87}^{\circ }$]

• A. $|\overrightarrow{R}|=17\text{unit}$ and ${43.26}^{\circ }$ North of East
• B. $|\overrightarrow{R}|=15.72\text{unit}$ and ${67.87}^{\circ }$ North of East
• C. $|\overrightarrow{R}|=15.72\text{unit}$ and ${67.87}^{\circ }$ angle between $\overrightarrow{A}\text{and}\overrightarrow{R}$
• D. $|\overrightarrow{R}|=15.72\text{unit}$ and ${67.87}^{\circ }$ angle between $\overrightarrow{B}\text{and}\overrightarrow{R}$

Answer 1

The correct option is B $|\overrightarrow{R}|=15.72\text{unit}$ and ${67.87}^{\circ }$ North of East

Given, $\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}$, $|\overrightarrow{A}|=15$ and $|\overrightarrow{B}|=10$
So, the given condition can be shown as

on resolving the vectors in the respective axes
$\overrightarrow{A}=\left(15\cos {30}^{\circ }\right)\hat{i}+\left(15\sin {30}^{\circ }\right)\hat{j}$
$\overrightarrow{A}=\left(12.99\right)\hat{i}+\left(7.5\right)\hat{j}$
$\overrightarrow{B}=(-10\cos {45}^{\circ })\hat{i}+\left(10\sin {45}^{\circ }\right)\hat{j}$
$\overrightarrow{B}=(-7.07)\hat{i}+\left(7.07\right)\hat{j}$
$\overrightarrow{R}=(12.99-7.07)\hat{i}+(7.5+7.07)\hat{j}$
$\overrightarrow{R}=\left(5.92\right)\hat{i}+\left(14.57\right)\hat{j}$
$|\overrightarrow{R}|=\sqrt{{5.92}^2+{14.57}^2}$
$|\overrightarrow{R}|=\sqrt{247.33}$
$|\overrightarrow{R}|=15.72\text{unit}$
direction of resultant $\theta ={\tan }^{-1}\left(\frac{14.57}{5.92}\right)={\tan }^{-1}\left(2.46\right)$
direction of resultant $\theta ={67.87}^{\circ }$ North of East.