# If the volume of a parallelopiped,

Question:

If the volume of a parallelopiped, whose coterminus edges are given by the vectors

$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\mathrm{n} \hat{\mathrm{k}}$

and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+n \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \quad(\mathrm{n} \geq 0)$, is $158 \mathrm{cu}$. units, then :

1. $\vec{a} \cdot \vec{c}=17$

2. $\vec{b} \cdot \vec{c}=10$

3. $n=7$

4. $\mathrm{n}=9$

Correct Option: , 3

Solution:

$v=[\vec{a} \vec{b} \vec{c}]$

$158=\left|\begin{array}{ccc}1 & 1 & n \\ 2 & 4 & -n \\ 1 & n & 3\end{array}\right|, n \geq 0$

$158=1\left(12+n^{2}\right)-(6+n)+n(2 n-4)$

$158=n^{2}+12-6-n+2 n^{2}-4 n$

$3 n^{2}-5 n-152=0$

$\mathrm{n}=8,-\frac{38}{6}$ (rejected)

$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=1+\mathrm{n}+3 \mathrm{n}=1+4 \mathrm{n}=33$

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