#L_a->(x+1=2t_1,y+1=3t_1,z+1=4t_1)#

#L_b->(x+1=3t_2,y=4t_2,z=5t_2)#

or

#L_a->p=p_a+t_1 vec v_a#

#L_b->p=p_b+t_2 vec v_b#

with

#p_a=(-1,-1,-1)# and #vec v_a=(2,3,4)#

#p_b=(-1,0,0)# and #vec v_b=(3,4,5)#

calling now #hat v_c = (vec v_a xx vec v_b)/norm(vec v_a xx vec v_b)# and also making

#Delta p = p_a+t_1 vec v_a-(p_b+t_2 vec v_b) = p_a-p_b+t_1 vec v_a-t_2 vec v_b#

making the scalar product by #hat v_c# we have

#<< Delta p, hat v_c >> = << p_a-p_b, hat v_c >># because #<< hat v_c, vec v_a >> = << hat v_c, vec v_b >> = 0# and finally

#d = abs(<< Delta p, hat v_c >>) = abs(<< p_a-p_b, hat v_c >>)#

In our case study we have

#p_a-p_b = (0,1,1)#

#vec v_a xx vec v_b = (-1,2,-1)# and

#hat v_c = (-1/sqrt[6], sqrt[2/3], -1/sqrt[6])# and

#d = sqrt[2/3] - 1/sqrt[6] = 0.408248#