Question

If $\left|\overline{a}\right|=3,\left|\overline{b}\right|=4$, angle between $\overline{a},\overline{b}is{120}^o$ then $|4\overline{a}+3\overline{b}|is{120}^o$ is equal to

• A. $25$
• B. $7$
• C. $13$
• D. $12$

Answer 1

The correct option is D $12$

$|\overrightarrow{a}|=3,|\overrightarrow{b}|=4$ and $\theta ={120}^o$

$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta =3.4\cos {120}^o=-6$

$|4\overrightarrow{a}+3\overrightarrow{b}{|}^2=(4\overrightarrow{a}+3\overrightarrow{b}).(4\overrightarrow{a}+3\overrightarrow{b})$

$=4\overrightarrow{a}.4\overrightarrow{a}+3\overrightarrow{b}.4\overrightarrow{a}+4\overrightarrow{a}.3\overrightarrow{b}+3\overrightarrow{b}.3\overrightarrow{b}$

$16|\overrightarrow{a}|+24\overrightarrow{a}\overrightarrow{b}+9|\overrightarrow{b}{|}^2[\because \overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{a}]$

$=16(3{)}^2+24(-6)+9(4{)}^2$

$=144-144+144=144$

$\therefore |4\overrightarrow{a}+3\overrightarrow{b}|=\sqrt{144}=12$