The angle between the pair of straight lines given by $\vec{r}=(3 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(\hat{i}+2 \hat{j}+2 \hat{k})$ and $\vec{r}=5 \hat{i}-2 \hat{j}+\mu(3 \hat{i}+2 \hat{j}+6 \hat{k})$ is:

$\cos ^{-1}\left(\frac{19}{21}\right)$

The correct answer is Option (1) - $\cos ^{-1}\left(\frac{19}{21}\right)$

angle between lines = angle between vectors parallel to lines

$\vec{b_1}=\hat i+2\hat j+2\hat k$

$\vec{b_2}=3\hat i+2\hat j+6\hat k$

$\cos θ=\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}=\frac{3+4+12}{\sqrt{1+4+4}\sqrt{9+4+36}}$

$=\frac{19}{3×7}$

$θ=\cos^{-1}\frac{19}{21}$