Question

Two lines \(L_1\) and \(L_2\) are given as \(L_1: \vec{r} = (\hat{i}+2\hat{j}+3\hat{k}) + \lambda(2\hat{i}+3\hat{j}+4\hat{k})\) and \(L_2: \vec{r} = (2\hat{i}+4\hat{j}+5\hat{k}) + \mu(4\hat{i}+6\hat{j}+8\hat{k})\), where \(\lambda\) and \(\mu\) are real parameters. Then which of the following statements are correct ?
A. Shortest distance between line \(L_1\) and \(L_2\) is \(\sqrt{\dfrac{5}{29}}\).
B. line \(L_1\) and \(L_2\) are parallel.
C. line \(L_1\) and \(L_2\) are concurrent lines.
D. line \(L_1\) and \(L_2\) are perpendicular.
Choose the correct answer from the options given below:

• A. B and C only
• B. C and D only
• C. A and C only
• D. A and B only

Hint:

Note \(\vec b_2=2\vec b_1\), so the lines are parallel. Use \(|(\vec a_2-\vec a_1)\times\vec b_1|/|\vec b_1|\) for the distance.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Two lines \(\vec r=\vec a_1+\lambda\vec b_1\) and \(\vec r=\vec a_2+\mu\vec b_2\) can be parallel, intersecting (concurrent) or skew. We test this using the direction vectors and the vector between the two given points.

Step 2: Read the data.
\(\vec a_1 = (1,2,3)\), \(\vec b_1=(2,3,4)\).
\(\vec a_2=(2,4,5)\), \(\vec b_2=(4,6,8)\).
Notice \(\vec b_2 = 2\vec b_1\).

Step 3: Check statement (B).
Since \(\vec b_2=2\vec b_1\), the direction vectors are proportional. So the lines are parallel. (B) is TRUE.

Step 4: Check statement (D).
The dot product is \(\vec b_1\cdot\vec b_2 = 8+18+32 = 58 \neq 0\). So the lines are not perpendicular. (D) is FALSE.

Step 5: Check statement (A).
For parallel lines the distance is \(\dfrac{|(\vec a_2-\vec a_1)\times \vec b_1|}{|\vec b_1|}\).
\(\vec a_2-\vec a_1=(1,2,2)\).
\[ (1,2,2)\times(2,3,4) = (2\cdot4-2\cdot3,\ 2\cdot2-1\cdot4,\ 1\cdot3-2\cdot2) = (2,0,-1) \]
Its magnitude is \(\sqrt{4+0+1}=\sqrt5\). Also \(|\vec b_1|=\sqrt{4+9+16}=\sqrt{29}\).
\[ d=\frac{\sqrt5}{\sqrt{29}}=\sqrt{\frac{5}{29}} \]
So (A) is TRUE.

Step 6: Check statement (C).
Concurrent lines meet at a point. Our lines are parallel and the distance between them is not zero, so they never meet. (C) is FALSE.

Step 7: Match with the options.
A and B are correct. That is option 4. Options 1, 2 and 3 all include C, which is false.

Final Answer:
Statements A and B are correct, which is option 4. \[ \boxed{\text{A and B only (Option 4)}} \]