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

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

The correct answer is Option (2) → $\cos^{-1}\left(\frac{19}{21}\right)$ ##

Given lines are $\mathbf{r} = 3\hat{i} - 2\hat{j} + 6\hat{k} + \lambda(2\hat{i} + \hat{j} + 2\hat{k})$

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

On comparing the given equation with $\mathbf{r} = \mathbf{a_1} + \lambda \mathbf{b_1}$ and $\mathbf{r} = \mathbf{a_2} + \lambda \mathbf{b_2}$, we get

$\mathbf{a_1} = 3\hat{i} - 2\hat{j} + 6\hat{k}, \mathbf{b_1} = 2\hat{i} + \hat{j} + 2\hat{k}$

and $\mathbf{a_2} = 2\hat{j} - 5\hat{k}, \mathbf{b_2} = 6\hat{i} + 3\hat{j} + 2\hat{k}$

If $\theta$ is angle between the given lines, then

$\cos \theta = \frac{|\mathbf{b_1} \cdot \mathbf{b_2}|}{|\mathbf{b_1}| \cdot |\mathbf{b_2}|}$

$= \frac{|(2\hat{i} + \hat{j} + 2\hat{k}) \cdot (6\hat{i} + 3\hat{j} + 2\hat{k})|}{|2\hat{i} + \hat{j} + 2\hat{k}||6\hat{i} + 3\hat{j} + 2\hat{k}|}$

$= \frac{|12 + 3 + 4|}{\sqrt{9}\sqrt{49}} = \frac{19}{21}$

$∴\theta = \cos^{-1} \frac{19}{21}$