Question

Let the volume of a parallelepiped 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 $\overrightarrow{w}=2\hat{i}+\hat{j}+\hat{k}$ be $1$ cu. unit. If $\theta$ be the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w},$ then the value of $cos\theta$ can be

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

Answer 1

Answer: (B)

$\pm 1=\left|\begin{array}{ccc}1& 1& \lambda \\ 1& 1& 3\\ 2& 1& 1\end{array}\right|\Rightarrow =-\lambda +3=\pm 1\Rightarrow \lambda =2or\lambda =4$
For $\lambda =4$
$cos\theta =\frac{2+1+4}{\sqrt{6}\sqrt{18}}=\frac{7}{6\sqrt{3}}$