Question

prove that a line through A(0,-1,-1) and coordinate of B(4,5,1) intersect the line through C(3,9,4) and D(-4,4,4)

Answer 1

I will use the parametric representation of a line in 3-dimensions using a vector and a point on the line,  to solve this.  There could be other ways.

     vector AB = 4 i + 6 j + 2 k             vector  CD = 7 i - 5 j + 0 k

parametric representation of line AB with point A (0, -1, -1) on it :
          x = 0 + 4 t ,     y = -1 + 6 t    ,        z = -1 + 2 t
That of line CD is with point C is,
       x = 3 + 7 s          y = 9 - 5 s            z  = 4 + 0 s
Solve these pairs of equations to get t and s.
       s = -1  ,  t = 5/2

Point of intersection is    (10, 14, 4)