Question

Find the angle between the pair of lines given by
$\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ and $\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$.

Answer 1

$\overrightarrow{r_1}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ is parallel to $(\hat{i}+2\hat{j}+2\hat{k})$

DR's $(1,2,2)$

$\overrightarrow{r_2}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$ is parallel to $(3\hat{i}+2\hat{j}+6\hat{k})$

DR's $(3,2,6)$

Angle between $\overrightarrow{r_1}$ and $\overrightarrow{r_1}$ is angle between DR's.

$\cos \theta =\frac{3\times 1+2\times 2+6\times 2}{\sqrt{1^2+2^2+2^2}\sqrt{3^2+2^2+6^2}}=\frac{19}{21}$

$\theta ={\cos }^{-1}\frac{19}{21}$