Question

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

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

Answer 1

Answer: (B)

$v =[\vec{ a }\, \vec{ b }\, \vec{ c }]$
$158 =\begin{vmatrix}1&1&n\\ 2&4&-n\\ 1&n&3\end{vmatrix}, n \ge\,0$
$158=1\left(12+ n ^{2}\right)-(6+ n )+ n (2 n -4)$
$158= n ^{2}+12-6- n +2 n ^{2}-4 n$
$3 n ^{2}-5 n -152=0$
$n =8,-\frac{38}{6}$ (rejected)
$\vec{ a } \cdot \vec{ c }=1+ n +3 n =1+4 n =33$
$\vec{ b } \cdot \vec{ c }=2+4 n -3 n =2+ n =10$