Question

If $a$ and $b$ are the two vectors such that $\left|a\right|=3\sqrt{3},\left|b\right|=4$ and $|a+b|=\sqrt{7}$, then the angle between $a$ and $b$ is

• A. ${120}^o$
• B. ${60}^o$
• C. ${30}^o$
• D. ${150}^o$

Answer 1

The correct option is D ${150}^o$

Given that, $\left|a\right|=3\sqrt{3},\left|b\right|=4$ and $|a+b|=\sqrt{7}$
Now, ${|a+b|}^2={\left|a\right|}^2+{\left|b\right|}^2+2\left|a\right|\left|b\right|\cos \theta$
$\Rightarrow {\left(\sqrt{7}\right)}^2={\left(3\sqrt{3}\right)}^2+4^2+2\left(3\sqrt{3}\right)\left(4\right)\cos \theta$
$\Rightarrow 7=27+16+24\sqrt{3}\cos \theta$
$\Rightarrow 24\sqrt{3}\cos \theta =-36$
$\Rightarrow \cos \theta =-\frac{3}{2\sqrt{3}}=-\frac{\sqrt{3}}{2}$
$\Rightarrow \theta ={150}^o$