Question

Volume of parallelopiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k},\overrightarrow{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\overrightarrow{w}=2\hat{i}+\hat{j}+\hat{k}$ is $1$ cu. unit. If $\theta$ be the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$, then $\cos \theta$ can be:

• A. $\frac{7}{6\sqrt{6}}$
• B. $\frac{5}{7}$
• C. $\frac{7}{6\sqrt{3}}$
• D. $\frac{5}{3\sqrt{3}}$

Answer 1

The correct option is C $\frac{7}{6\sqrt{3}}$

Volume of parallelepiped $=\left[\overrightarrow{u}\overrightarrow{v}\overrightarrow{w}\right]$

$\Rightarrow \left|\begin{array}{ccc}1& 1& \lambda \\ 1& 1& 3\\ 2& 1& 1\end{array}\right|=\pm 1$

$\Rightarrow \lambda =2$ or $4$
For $\lambda =4,$
$\cos \theta =\frac{2+1+4}{\sqrt{18}\sqrt{6}}=\frac{7}{6\sqrt{3}}$