Question

If the volume of a parallelopiped, whose coterminuous edges are given by the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+n\hat{k},\overrightarrow{b}=2\hat{i}+4\hat{j}-n\hat{k}$ and $\overrightarrow{c}=\hat{i}+n\hat{j}+3\hat{k}(n\ge 0)$, is $158$ cu.units, then:

• A. $\overrightarrow{a}\cdot \overrightarrow{c}=17$
  
• B. $\overrightarrow{b}\cdot \overrightarrow{c}=10$
• C. $n=9$
  
• D. $n=7$
  

Answer 1

The correct option is B $\overrightarrow{b}\cdot \overrightarrow{c}=10$

$\left|\begin{array}{ccc}1& 1& n\\ 2& 4& -n\\ 1& n& 3\end{array}\right|=158$
$\Rightarrow (12+n^2)-(6+n)+n(2n-4)=158$
$\Rightarrow 3n^2-5n+6-158=0$
$\Rightarrow 3n^2-5n-152=0$
$\Rightarrow 3n^2-24n+19n-152=0$
$\Rightarrow (3n+19)(n-8)=0$
$\Rightarrow n=8$

$\therefore \overrightarrow{b}\cdot \overrightarrow{c}=2+4n-3n=2+n=10$