Question

The resultant of three vectors $1,2,$ and $3$ units whose directions are those of the sides of an equilateral triangle is at an angle of

• A. $30$ with the first vector
• B. $15$ with the first vector
• C. $100$ with the first vector
• D. $150$ with the first vector

Answer 1

Answer: (D)

$150$ with the first vector

Consider the problem

Let $\overrightarrow{R}$ be resultant vector

$\overrightarrow{R}=[1-(2\sin {30}^{\circ }+3\sin {30}^{\circ }\left)\right]\overrightarrow{i}+(2\cos {30}^{\circ }-3cos{30}^{\circ })\overrightarrow{j}$

$\overrightarrow{R}=[1-(1+1.5\left)\right]\overrightarrow{i}-\frac{\sqrt{3}}{2}\overrightarrow{j}$

$\overrightarrow{R}=-[\frac{3}{2}\overrightarrow{i}+\frac{\sqrt{3}}{2}\overrightarrow{j}]$

$\tan \theta =\frac{\frac{\sqrt{3}}{2}}{\frac{3}{2}}=\frac{1}{\sqrt{3}}$

Therefore,

$\theta ={30}^{\circ }$

Angle from $1$ vector $=180-30=150$ (clockwise sense)

Hence option $D$ is the correct answer.