Mathematics - Vector Question with Solution | TestHub

Given $\overrightarrow{a}+2\overrightarrow{b}+2\overrightarrow{c}=0$

$\Rightarrow \overrightarrow{a}+2\overrightarrow{c}=-2\overrightarrow{b}$

Squaring both sides,

$\Rightarrow {\left|\overrightarrow{a}\right|}^2+4{\left|\overrightarrow{c}\right|}^2+4\overrightarrow{a}.\overrightarrow{c}=4{\left|\overrightarrow{b}\right|}^2$

$\Rightarrow 1+4+4\cos \theta =4\Rightarrow \cos \theta =-\frac{1}{4}$

$\therefore \sin \theta =\sqrt{1-{\cos }^2\theta }=\frac{\sqrt{15}}{4}$

Now, $\left|\overrightarrow{a}\times \overrightarrow{c}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{c}\right|\sin \theta =\frac{\sqrt{15}}{4}$