Question

The resultant of the two vectors makes angles ${30}^{\circ }$ and ${60}^{\circ }$ with them and the magnitude of the resultant is $40$. The magnitudes of two vectors are

• A. $20$ and $40$
• B. $20$ and $20\sqrt{3}$
• C. $20$ and $25$
• D. $10\sqrt{3}$ and $20$

Answer 1

The correct option is B $20$ and $20\sqrt{3}$

Let $\overrightarrow{R}$ be the resultant and makes ${30}^{\circ }$ with$\overrightarrow{Q}$ and ${60}^{\circ }$ with $\overrightarrow{P}$ as shown in figure.

from the diagram,
Taking $\alpha$ angle between $\overrightarrow{P}$ and $\overrightarrow{R}$
$\alpha ={60}^{\circ }$$\tan \alpha =\frac{Q\sin {90}^{\circ }}{P+Q\cos {90}^{\circ }}$
$\tan {60}^{\circ }=\frac{Q\sin {90}^{\circ }}{P+Q\cos {90}^{\circ }}$
$\Rightarrow \sqrt{3}=\frac{Q}{P}$

$\Rightarrow Q=\sqrt{3}P$
And also,
$R=\sqrt{P^2+Q^2}$
$\Rightarrow 40=\sqrt{P^2+Q^2}$
$\Rightarrow 40=\sqrt{P^2+3P^2}$
$\Rightarrow P=20$ and $Q=20\sqrt{3}$