Vectors and the Geometry of Space

You failed to include equations 1 and 2.

 

Suppose line 1 is R₁(t) = P + tD₁
Line 2 is R₂(s) = Q + sD₂

where the P and Q are given points and the D's are given direction vectors.

Then a line from one to the other would be
R₃(u) = R₁(t) + u(R₂(s)-R₁(t))

Set that equal to your given point. You have three given coordinates of the point and three parameters s,t, and u to work with.

 

The line R₃ goes from the line with parameter t to the other with parameter s with t and s unknown. You have to choose s and t so that it's possible for R₃ to go through A for some value of u. For most choices of s and t the line won't pass through A for any value of u.

[Edit -- Added] The general formula you get is long and messy but doing it for specific lines and points isn't too difficult.