<< Calculus`VectorAnalysis`

A = {1, 1, 1};
B = {-1, 1, -1};
V = {-1, -1, - .2};
W = {1 , -1.2, - .8};

LV = Line[{A, A + V, A + 2  V}]
LW = Line[{B, B + W, B + 2  W}]
Show[Graphics3D[LV], Graphics3D[LW], Axes -> True]

s = ((A . W - B . W) (W . V) + (B . V - A . V) (W . 
                  W))/((V . V) (W . W) - (V . W) (W . 
                  V))
t = ((B . V - A . V) (V . W) + (A . W - B . W) (V . 
                  V))/((W . W) (V . V) - (V . W) (W . 
                  V))

P = Point[A + s*V]
Q = Point[B + t*W]

Show[Graphics3D[LV], Graphics3D[LW], Graphics3D[{PointSize[0.02], P}], 
      Graphics3D[{PointSize[0.02], Q}], ViewPoint -> {-2, 1, 1}, 
      Axes -> True]

p = A + s*V
q = B + t*W
U = q - p

U . V
U . W