Question

\(\text{The distance between the lines } \vec{r} = \hat{i} - 2\hat{j} + 3\hat{k} + \lambda (2\hat{i} + 3\hat{j} + 6\hat{k}) \text{ and } \vec{r} = 3\hat{i} - 2\hat{j} + \hat{k} + \mu (4\hat{i} + 6\hat{j} + 12\hat{k}) \text{ is:}\)

• A. \( \frac{\sqrt{28}}{7} \)
• B. \( \frac{\sqrt{199}}{7} \)
• C. \( \frac{\sqrt{328}}{7} \)
• D. \( \frac{\sqrt{421}}{7} \)

Answer 1

The Correct Option is C

Use the formula for the distance between two skew lines:

\[ d = \frac{|(\vec{d}_1 \times \vec{d}_2) \cdot (\vec{r}_2 - \vec{r}_1)|}{|\vec{d}_1 \times \vec{d}_2|}. \]

Substitute the direction vectors and points from the lines and calculate the cross product and dot product as required.

Simplify to confirm that the distance is \(\frac{\sqrt{328}}{7}\), verifying option (3) as the correct answer.