Question

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

Answer 1

Given lines are

$\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ and
$\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$

The direction ratios of the lines are $(1,2,2)$ and $(3,2,6)$

Then angle between the lines is given by

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

$=\frac{19}{\sqrt{9}\sqrt{49}}$

$=\frac{19}{21}$

$\therefore$ $\cos \theta =\frac{19}{21}$

Hence, $\theta ={\cos }^{-1}\left(\frac{19}{21}\right)$