Question

The length of the shortest distance between the lines \( \mathbf{r} = 3i + 5j + 7k + \lambda(2i - 2j + 3k) \) and \( \mathbf{r} = -i - j + k + \mu(7i - 6j + k) \) is:

• A. \( 83 \, \text{units} \)
• B. \( \sqrt{6} \, \text{units} \)
• C. \( \sqrt{3} \, \text{units} \)
• D. \( \sqrt{29} \, \text{units} \)

Hint:

The shortest distance between skew lines is calculated using the vector cross product and the position vectors of points on the lines.

Answer 1

The Correct Option is D

Step 1: Shortest distance between skew lines. The shortest distance \( d \) between two skew lines is given by the formula: \[ d = \frac{|(\mathbf{b}_1 - \mathbf{b}_2) \cdot (\mathbf{n}_1 \times \mathbf{n}_2)|}{|\mathbf{n}_1 \times \mathbf{n}_2|} \] Where \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \) are points on the lines and \( \mathbf{n}_1 \), \( \mathbf{n}_2 \) are the direction vectors of the lines.
Step 2: Conclusion. The shortest distance between the lines is \( \sqrt{29} \, \text{units} \).