Question

A parallelogram is constructed on the vectors $a=3\alpha -\beta ,b=\alpha +3\beta ,$ if $|\alpha |=|\beta |=2$ and angle between $\alpha$ and $\beta$ is $\frac{\pi }{3},$ then length of a diagonal of the parallelogram is

• A. $4\sqrt{5}$
• B. $4\sqrt{3}$
• C. $4\sqrt{17}$
• D. None of these

Answer 1

The correct option is D None of these

$\overrightarrow{a}$ and $\overrightarrow{b}$ are two adjacent sides of ll gm so sum of sides (adjacent)

will give length of diagonal.

$a=3\alpha -\beta$ of $\alpha +3\beta$

$a+b=4\alpha +2\beta$

$|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{(4\alpha {)}^2+(2\beta {)}^2+2\left(4\alpha \right)\cdot \left(2\beta \right)}$

$=\sqrt{{16|\alpha |}^2+4|\beta {|}^2+16\alpha \cdot \beta }$

$=\sqrt{16\left(4\right)+4\left(4\right)+16|\alpha ||\beta |\cos \theta }$

where $\theta$ is angle between $\alpha \&\beta$

$\begin{array}{cc}|\overrightarrow{a}+\overrightarrow{b}|& =\sqrt{64+16+64\cos \frac{\pi }{3}}\\ & =\sqrt{64+16+32}\\ & =\sqrt{112}=4\sqrt{7}\end{array}$

so required length of diagonal is $4\sqrt{7}$.

Answer : option (D)