Question

Let the volume of a parallelepiped whose coterminous edges are given by $\overset{ \rightarrow }{u}=\hat{i}+\hat{j}+\lambda \hat{k},\overset{ \rightarrow }{ \, v}=\hat{i}+\hat{j}+3\hat{k}$ and $\overset{ \rightarrow }{w}=2\hat{i}+\hat{j}+\hat{k}$ be $1$ cu. unit. If $\theta $ be the angle between the edges $\overset{ \rightarrow }{u}$ and $\overset{ \rightarrow }{w},$ 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)

$\pm1=\begin{vmatrix} 1 & 1 & \lambda \\ 1 & 1 & 3 \\ 2 & 1 & 1 \end{vmatrix}\Rightarrow =-\lambda +3=\pm1\Rightarrow \lambda =2or\lambda =4$
For $\lambda =4$
$cos\theta =\frac{2 + 1 + 4}{\sqrt{6} \sqrt{18}}=\frac{7}{6 \sqrt{3}}$