Question

The ratio of maximum and minimum magnitudes of the resultant of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $3:1$. Now, $|\overrightarrow{a}|$ is equal to

• A. $|\overrightarrow{b}|$
• B. $2|\overrightarrow{b}|$
• C. $3|\overrightarrow{b}|$
• D. $4|\overrightarrow{b}|$

Answer 1

Answer: (B)

$2|\overrightarrow{b}|$

Resultant of two vectors is mark when angle b/w them is 0. Resultant is min when angle is $180$.

$\therefore$ $\frac{|\overrightarrow{a}+\overrightarrow{b}|}{|\overrightarrow{a}-\overrightarrow{b}|}=\sqrt{\frac{a^2+b^2+2ab}{a^2+b^2-2ab}}=\frac{3}{1}$

$\Rightarrow a^2+b^2+2ab={9a}^2+{9b}^2-18ab$

$\Rightarrow {8a}^2+{8b}^2-20ab=0$

$\Rightarrow {2a}^2+{2b}^2-5ab=0$

Let $a=x.b$

$\Rightarrow {2x}^2{b}^2+{2b}^2-{5ab}^2=0$

$\Rightarrow {2x}^2+2-5x=0$

$\Rightarrow {2x}^2-5x+2=0$

$\Rightarrow {2x}^2-4x-x+2=0$

$\Rightarrow 2x(x-2)-1(x-2)=0$

$\Rightarrow (2x-1)(x-2)=0$

$\therefore$ $x=2$ or $x=\frac{1}{2}$

$\Rightarrow a=2b$