Question #ad27d

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Answer 1 · 2016-12-12T19:08:32

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

$L_a->(x+1=2t_1,y+1=3t_1,z+1=4t_1)$
$L_b->(x+1=3t_2,y=4t_2,z=5t_2)$

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

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