The angle between the lines $\bar{r}=(\hat{\imath}+2 \hat{\jmath}+3…
The angle between the lines $\bar{r}=(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})+\lambda(\hat{\imath}+\hat{\jmath}+2 \hat{k})$ and $\bar{r}=(3 \hat{\imath}+\hat{k})+\lambda^{\prime}(2 \hat{\imath}+\hat{\jmath}-\hat{k}), \lambda, \lambda^{\prime} \in \mathrm{R}$ is
$\cos ^{-1}\left(\frac{1}{6}\right)$
$\cos ^{-1}\left(\frac{1}{5}\right)$
$\cos ^{-1}\left(\frac{1}{3}\right)$
$\cos ^{-1}\left(\frac{2}{3}\right)$
Solution
The direction ratios of the lines are $1,1,2$ and $2,1,-1$ and let $\theta$ be the angle between them
$\begin{aligned}
\cos \theta &=\left|\frac{(1)(2)+(1)(1)+2(-1)}{\sqrt{1+1+4} \cdot \sqrt{4+1+1}}\right| \\
\cos \theta &=\left|\frac{1}{\sqrt{6} \cdot \sqrt{6}}\right|=\frac{1}{6} \Rightarrow \theta=\cos ^{-1}\left(\frac{1}{6}\right)
\end{aligned}$