Question

The shortest distance between the lines \[ \vec{r} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k} \] and \[ \vec{r} = (p + 1)\hat{i} + (2p - 1)\hat{j} + (2p + 1)\hat{k} \] is

• A. \( \dfrac{8}{\sqrt{29}} \) units
• B. \( \dfrac{4}{\sqrt{29}} \) units
• C. \( \dfrac{2}{\sqrt{5}} \) units
• D. \( \dfrac{4}{\sqrt{19}} \) units

Hint:

For skew lines, always use the vector triple product formula to compute the shortest distance accurately.

Answer 1

The Correct Option is C

Step 1: Write the direction vectors of the given lines.
From the first line, the direction vector is \[ \vec{d}_1 = -\hat{i} + \hat{j} - 2\hat{k} \] From the second line, the direction vector is \[ \vec{d}_2 = \hat{i} + 2\hat{j} + 2\hat{k} \]

Step 2: Find a point on each line.
For the first line, taking \( t = 0 \), a point is \[ A(1, -2, 3) \] For the second line, taking \( p = 0 \), a point is \[ B(1, -1, 1) \]

Step 3: Use the formula for shortest distance between two skew lines.
\[ \text{Shortest distance} = \frac{|(\vec{AB} \cdot (\vec{d}_1 \times \vec{d}_2))|}{|\vec{d}_1 \times \vec{d}_2|} \] After calculation, \[ \text{Shortest distance} = \frac{2}{\sqrt{5}} \]

Step 4: Conclusion.
Hence, the shortest distance between the given lines is \[ \boxed{\dfrac{2}{\sqrt{5}}} \]