How can you intersect a cube at a single point?

Geometry Level 4

In the \( (x, y, z, w) \) coordinate system, there is a line through the points \( (1, 3, -8, 4) \) and \( (9, 8, 5, 6) \). If the point of intersection of the line with the \(zwx\) 3D space/cube/whatever you call it (point, line, plane, ??? ) can be represented as the point \( \left( -\dfrac{a}{b}, \dfrac{c}{d}, -\dfrac{e}{f}, \dfrac{g}{h} \right) \), where each pair in parentheses \( (a, b), (e, f), (g, h) \) contain only coprime positive integers, find the minimum sum of \( a+b+c+d+e+f+g+h \).

Bonus: Make sense of the problem from a geometric point of view.

Notes and Assumptions:

\( e \) here is a variable and isn't equal to \( \displaystyle \lim_{n \to \infty} \left( 1 + \dfrac{1}{n} \right)^n \) or \( \displaystyle \sum_{k=0}^{\infty} \dfrac{1}{k!} \)

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