To find the shortest distance between two lines in 3-dimensional space, we need the equations of both lines. From the prompt, it seems we have the equation for only one line, which is given in a symmetrical form:
\[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \]
Let's assume that is Line 1. To proceed with finding the shortest distance, we need the equation for the second line, Line 2, which is not provided. Without the equation for the second line, we cannot calculate the shortest distance between them.
However, if the second line were given in the form:
\[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \]
where \( (x_0, y_0, z_0) \) is a point on Line 2 and \( a, b, c \) correspond to the direction ratios of Line 2, then we could find the shortest distance as follows:
1. Make sure that the lines are not parallel. Parallel lines will either not have a unique shortest distance (if they're identical) or the shortest distance will be equivalent to the distance from any point on one line to the other if they're strictly parallel.
2. If the lines are not parallel, find a vector that is perpendicular to both lines, which can be done by finding the cross product of the direction vectors of the two lines.
3. Calculate the distance between a specific point on Line 1 and Line 2 using this perpendicular vector.
If we denote \( \vec{d_1} \) to be the direction vector of Line 1 and \( \vec{d_2} \) to be the direction vector of Line 2, and \( \vec{p_1} \) to be a point on Line 1 and \( \vec{p_2} \) to be a point on Line 2, then the distance \( D \) can be calculated with the formula:
\[ D = \frac{|\vec{d_1} \times \vec{d_2} \cdot (\vec{p_1} - \vec{p_2})|}{|\vec{d_1} \times \vec{d_2}|} \]
But since we do not have the second line's equation, we cannot perform this calculation at the moment. If you could provide the equation for the second line, we could certainly go ahead and find the shortest distance between them.