Question

The angle between the lines \(\overrightarrow{r}=\hat{i}+4\hat{k}+\lambda(2\hat{i}+\hat{j}-\hat{k})\) and \(\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}+\mu(3\hat{i}+\hat{k})\) is

• A. \(\cos^{-1}(\frac{\sqrt{5}}{6})\)
• B. \(\cos^{-1}(\frac{\sqrt{15}}{6})\)
• C. \(\cos^{-1}(\frac{1}{12})\)
• D. \(\cos^{-1}(\frac{\sqrt{15}}{15})\)
• E. \(\cos^{-1}(\frac{\sqrt{3}}{30})\)

Answer 1

The Correct Option is B

The given equations of the lines are: 

\[ \overrightarrow{r} = \hat{i} + 4\hat{k} + \lambda(2\hat{i} + \hat{j} - \hat{k}) \] \[ \overrightarrow{r} = 2\hat{i} - \hat{j} + 3\hat{k} + \mu(3\hat{i} + \hat{k}) \]

Direction vectors:

\[ \overrightarrow{d_1} = 2\hat{i} + \hat{j} - \hat{k} \] \[ \overrightarrow{d_2} = 3\hat{i} + 0\hat{j} + \hat{k} \]

Formula for angle between two lines:

\[ \cos\theta = \frac{\overrightarrow{d_1} \cdot \overrightarrow{d_2}}{|\overrightarrow{d_1}| |\overrightarrow{d_2}|} \]

Computing the dot product:

\[ (2\hat{i} + \hat{j} - \hat{k}) \cdot (3\hat{i} + 0\hat{j} + \hat{k}) \] \[ = (2 \times 3) + (1 \times 0) + (-1 \times 1) = 6 + 0 - 1 = 5 \]

Computing magnitudes:

\[ |\overrightarrow{d_1}| = \sqrt{2^2 + 1^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \] \[ |\overrightarrow{d_2}| = \sqrt{3^2 + 0^2 + 1^2} = \sqrt{9 + 0 + 1} = \sqrt{10} \]

Substituting values:

\[ \cos\theta = \frac{5}{\sqrt{6} \times \sqrt{10}} \] \[ = \frac{5}{\sqrt{60}} = \frac{5}{2\sqrt{15}} = \frac{\sqrt{15}}{6} \]

Final Answer:

\[ \cos^{-1} \left(\frac{\sqrt{15}}{6} \right) \]