Question

If the shortest distance between the lines.
L1: $\vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}$, $\lambda \in \mathbb{R}$.
L2: $\vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}$, $\mu \in \mathbb{R}$ is $\frac{m}{\sqrt{n}}$, where gcd(m, n) = 1, then the value of m + n equals.

• A. 384
• B. 387
• C. 377
• D. 390

Answer 1

The Correct Option is B

The shortest distance between skew lines is given by:

\[ \text{Shortest Distance} = \frac{| \mathbf{AB} \cdot (\mathbf{p} \times \mathbf{q}) |}{|\mathbf{p} \times \mathbf{q}|}. \]

Step 1: Input values:

\[ \mathbf{p} = \begin{bmatrix} 1 \\ -3 \\ 4 \end{bmatrix}, \quad \mathbf{q} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{AB} = \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}. \]

Step 2: Compute \(\mathbf{p} \times \mathbf{q}\):

\[ \mathbf{p} \times \mathbf{q} = \begin{bmatrix} -4 \\ -3 \\ 4 \end{bmatrix}. \]

Magnitude of \(\mathbf{p} \times \mathbf{q}\):

\[ |\mathbf{p} \times \mathbf{q}| = \sqrt{(-4)^2 + (-3)^2 + 4^2} = \sqrt{55}. \]

Step 3: Calculate \(|\mathbf{AB} \cdot (\mathbf{p} \times \mathbf{q})|\):

\[ \mathbf{AB} \cdot (\mathbf{p} \times \mathbf{q}) = (0)(-4) + (2)(-3) + (2)(4) = -6 + 8 = 2. \] \[ |\mathbf{AB} \cdot (\mathbf{p} \times \mathbf{q})| = 32. \]

Step 4: Shortest Distance:

\[ \text{Shortest Distance} = \frac{32}{\sqrt{355}}. \]

Step 5: Simplify:

\[ m = 32, \quad n = 355, \quad \gcd(m, n) = 1. \]

Sum:

\[ m + n = 32 + 355 = 387. \]

Final Answer:

\[ \boxed{387.} \]