Mathematics - Vector Question with Solution | TestHub

Let the angle between$\overrightarrow{a}$and$\overrightarrow{c}$is$\theta .$

$\therefore \cos \theta =\frac{\overrightarrow{a} . \overrightarrow{c}}{\left|\overrightarrow{a}\right| \left|\overrightarrow{c}\right|}=\frac{4}{\sqrt{2} \left|\overrightarrow{c}\right|}$

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

$\Rightarrow \left|\overrightarrow{a}\times \overrightarrow{c}\right|=\left|\overrightarrow{b}\right|$

$\Rightarrow \left|\overrightarrow{a}\right|. \left|\overrightarrow{c}\right|\sin \theta =\left|\overrightarrow{b}\right|$

$\Rightarrow {\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{c}\right|}^2{\sin }^2\theta ={\left|\overrightarrow{b}\right|}^2$

$\Rightarrow 2{\left|\overrightarrow{c}\right|}^2\left(1-{\cos }^2\theta \right)=3$

$\Rightarrow 2{\left|\overrightarrow{c}\right|}^2\left(1-\frac{16}{2{\left|\overrightarrow{c}\right|}^2}\right)=3$

$\Rightarrow 2{\left|\overrightarrow{c}\right|}^2-16=3$

$\Rightarrow {\left|\overrightarrow{c}\right|}^2=\frac{19}{2}$