Question

The shortest distance between the lines \(\overset{⃗}{r} = (4\hat{i}-\hat{j})+λ(\hat{i}+2\hat{j}-3\hat{k})\) and \(\overset{⃗}{r} = (\hat{i}-\hat{j}+2\hat{k})+μ(\hat{i}+4\hat{j}-5\hat{k})\) is...

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

Hint:

The shortest distance between skew lines is |(a2 - a1) . (d1 x d2)| / |d1 x d2|.

Answer 1

The Correct Option is C

Step 1: Direct idea
The distance equals the length of the projection of the vector joining the two lines on the common perpendicular direction $\hat{n}$.

Step 2: Common perpendicular
$\vec{n} = \vec{d}_1\times\vec{d}_2 = (2, 2, 2)$, giving the unit vector $\hat{n} = \dfrac{1}{\sqrt{3}}(1, 1, 1)$.

Step 3: Project
$(\vec{a}_2 - \vec{a}_1)\cdot\hat{n} = \dfrac{-3 + 0 + 2}{\sqrt{3}} = -\dfrac{1}{\sqrt{3}}$.

Step 4: Distance
The distance is the absolute value, $\dfrac{1}{\sqrt{3}}$. It is not zero, so the lines are skew.

Final Answer:
The shortest distance is 1/sqrt 3. This is option (C). \[ \boxed{\text{(C) }\frac{1}{\sqrt{3}}} \]