How can you find out if two lines expressed in their vector form intersect?

Let the lines be:

r₁ = (a, b, c) + t(d, e, f)

r₂ = (g, h, i) + s(j, k, l)

Since t and s above are variables and the rest of the letters are constants, the only way to change the point which the vector equations are referring to is by varying t and s. If the lines intersect, there must be some value of t and some value of s that results in r₁ equalling r₂. If there is no such point, the lines are skew (they do not intersect). The way to find the relevant values of t and s is simultaneously. Split up each equation above into 3 parts: x, y and z.

r₁:    x = a + d * t    y = b + e * t    z = c + f * t

r₂:    x = g + j * s    y = h + k * s    z = i + l * s

Therefore: a + d * t = g + j * s

    b + e * t = h + k * s

    c + f * t = i + l * s

Solve the first two equations simultaneously to find the values of t and s. Substitute them into the third equation. If a contradiction results, the lines are skew. If the third equation works with those values of t and s, the lines meet.