Question

The shortest distance between the lines \( x = y + 2 = 6z - 6 \) and \( x + 1 = 2y = -12z \) is:

• A. \( \frac{1}{2} \)
• B. 2
• C. 1
• D. \( \frac{3}{2} \)

Hint:

To calculate the shortest distance between two skew lines, use the formula involving the cross product of their direction vectors and the difference of any two points on them.

Answer 1

The Correct Option is B

The given equations represent two skew lines. To calculate the shortest distance between them, we will use the formula: \[ d = \frac{| \mathbf{a_2} - \mathbf{a_1} \cdot \left( \mathbf{b_1} \times \mathbf{b_2} \right) |}{| \mathbf{b_1} \times \mathbf{b_2} |} \] Where: - \( \mathbf{a_1} \) and \( \mathbf{a_2} \) are points on the respective lines, - \( \mathbf{b_1} \) and \( \mathbf{b_2} \) are the direction vectors of the lines. 
Step 1: Write the parametric equations for both lines. For the first line \( x = y + 2 = 6z - 6 \), we can set: \[ \mathbf{r_1} = (t, t-2, \frac{t+6}{6}) \] For the second line \( x + 1 = 2y = -12z \), we can set: \[ \mathbf{r_2} = (s-1, \frac{s}{2}, -\frac{s}{12}) \] 
Step 2: Find the vectors \( \mathbf{a_1} - \mathbf{a_2} \) and \( \mathbf{b_1} \times \mathbf{b_2} \). By using the vector cross product and dot product, you will arrive at the shortest distance formula. 
Step 3: Calculate the value of \( d \). Using the formula, we find that the shortest distance is 2.