Mathematics - Vector Question with Solution | TestHub

Given,

$\overrightarrow{a}$and $\overrightarrow{b}$ be two vectors, where $\left|\overrightarrow{a}\right|=1,\left|\overrightarrow{b}\right|=4$ and $\overrightarrow{a}·\overrightarrow{b}=2$,

Now given, $\overrightarrow{c}=\left(2\overrightarrow{a}\times \overrightarrow{b}\right)-3\overrightarrow{b}$

So, $\overrightarrow{b}·\overrightarrow{c}=\overrightarrow{b}·\left(2\overrightarrow{a}\times \overrightarrow{b}\right)-3\overrightarrow{b}·\overrightarrow{b}$

$\Rightarrow \overrightarrow{b}·\overrightarrow{c}=-3|b{|}^2$ $\left\{\text{as} \left[\begin{array}{ccc}\overrightarrow{b}& 2\overrightarrow{a}& \overrightarrow{b}\end{array}\right]=0 \text{or} \overrightarrow{b}·\left(2\overrightarrow{a}\times \overrightarrow{b}\right)=0\right\}$

$\Rightarrow \overrightarrow{b}·\overrightarrow{c}=-48$