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If I have two lines $$ \eqalign{ & L_1 \left( t \right):p_1 + td_1 \cr & L_2 \left( q \right):p_2 + qd_2 \cr} $$ living in $\mathbb{R}^n$, there exists a classical formula to find the distance between them involving dot and cross products. The question is: can I deduce that formula only using calculus? (In this case, 2 variables) i.e., find the values such that the function $$ f\left( {t,q} \right) = \left \| L_1 (t) - L_2(t) \right \| = \left \| p_1 + td_1 - p_2 - qd_2 \right\| $$ reaches its minimum value.

Oh sorry; for simplicity, to have the natural cross product, just take $\mathbb{R}^3$.

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