Mathematics - Vector Question with Solution | TestHub

Given: $\overrightarrow{p}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$

$\Rightarrow \left(\overrightarrow{p}-\overrightarrow{c}\right)\times \overrightarrow{b}=0$

$\Rightarrow \left(\overrightarrow{p}-\overrightarrow{c}\right)\left|\right|\overrightarrow{b}$

$\Rightarrow \left(\overrightarrow{p}-\overrightarrow{c}\right)=k\overrightarrow{b}$

$\Rightarrow \overrightarrow{p}=k\overrightarrow{b}+\overrightarrow{c}$

$\Rightarrow \overrightarrow{p}=\left(4k+1\right)\hat{i}+\left(k-3\right)\hat{j}+\left(7k+4\right)\hat{k}$

Now, $\overrightarrow{p}.\overrightarrow{a}=0$

$\Rightarrow 3\left(4k+1\right)+\left(k-3\right)-2\left(7k+4\right)=0$

$\Rightarrow k=-8$

$\Rightarrow \overrightarrow{p}=-31\hat{i}-11\hat{j}-52\hat{k}$

$\Rightarrow \overrightarrow{p}.\left(\hat{i}-\hat{j}-\hat{k}\right)=-31+11+52$

$\Rightarrow \overrightarrow{p}.\left(\hat{i}-\hat{j}-\hat{k}\right)=32$