Step 1: Problem Definition:
The objective is to compute the minimum distance between two lines presented in Cartesian coordinates. The initial step involves classifying the relationship between the lines: are they parallel, intersecting, or skew?Step 2: Governing Principles:
A line in Cartesian form is expressed as $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. Its vector representation is $\vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda(a\hat{i} + b\hat{j} + c\hat{k})$.The direction ratios for the first line are $(a_1, b_1, c_1)$, and for the second line, they are $(a_2, b_2, c_2)$.Parallelism is established when direction ratios are proportional, meaning $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.For parallel lines, the shortest distance is the perpendicular distance from a point on one line to the other.If the lines are not parallel, they must be either intersecting or skew.Step 3: Detailed Analysis:
Let the first line be L1 and the second line be L2.
L1: $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$. Direction ratios: $\vec{d_1} = (2, 3, 4)$.L2: $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$. Direction ratios: $\vec{d_2} = (4, 6, 8)$.
Parallelism Check: Compare direction ratios.
\[ \frac{4}{2} = 2 \]\[ \frac{6}{3} = 2 \]\[ \frac{8}{4} = 2 \]Proportional direction vectors ($\vec{d_2} = 2\vec{d_1}$) indicate that the lines are parallel.
To distinguish between coincident and distinct parallel lines, verify if a point from L1 lies on L2.
A point on L1 is $P_1 = (1, 2, 3)$. Test $P_1$ against L2's equation.
Substitute $P_1$'s coordinates into L2's equation:
\[ \frac{1-2}{4} = \frac{-1}{4} \]\[ \frac{2-4}{6} = \frac{-2}{6} = \frac{-1}{3} \]\[ \frac{3-5}{8} = \frac{-2}{8} = \frac{-1}{4} \]As $\frac{-1}{4} eq \frac{-1}{3}$, $P_1$ is not on L2. Thus, the lines are distinct and parallel.
The shortest distance (D) between two parallel lines, $\vec{r} = \vec{a_1} + \lambda\vec{d}$ and $\vec{r} = \vec{a_2} + \mu\vec{d}$, is computed using:
\[ D = \frac{|(\vec{a_2} - \vec{a_1}) \times \vec{d}|}{|\vec{d}|} \]From the line equations:
$\vec{a_1} = \hat{i} + 2\hat{j} + 3\hat{k}$
$\vec{a_2} = 2\hat{i} + 4\hat{j} + 5\hat{k}$
$\vec{d} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ (using the simplified direction vector)
$\vec{a_2} - \vec{a_1} = (2-1)\hat{i} + (4-2)\hat{j} + (5-3)\hat{k} = \hat{i} + 2\hat{j} + 2\hat{k}$
Cross product calculation:
\[ (\vec{a_2} - \vec{a_1}) \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
1 & 2 & 2
2 & 3 & 4 \end{vmatrix} = \hat{i}(8-6) - \hat{j}(4-4) + \hat{k}(3-4) = 2\hat{i} - \hat{k} \]Magnitude of the cross product: $|2\hat{i} - \hat{k}| = \sqrt{2^2 + (-1)^2} = \sqrt{5}$.
Magnitude of the direction vector: $|\vec{d}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4+9+16} = \sqrt{29}$.
Distance: $D = \frac{\sqrt{5}}{\sqrt{29}} = \sqrt{\frac{5}{29}}$.