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

Answer: (B)

387

Answer 2

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:

$\boxed{387.}$