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Proof that the distance is the minimum distance

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Proof that the distance is the minimum distance

Perhaps it was obvious to you (geometrically),
but I think at least some words should be said on why
the number d that you found is the _minimum_ distance between points
on the two lines.

An analytic proof might be the following.
Denote by d^2(s, t) = | a-b + su - tv |^2
the distance between any chosen points on the two lines.
we can write
d^2(s,t) = | (a-b).n n + w + su - tv |^2
where w is the component of a-b not parallel to n.
The last three terms lie on the plane perpendicular to n,
so by the Pythagorean theorem

d^2(s,t) = | (a-b).n n |^2 + |w + su - tv|^2

since the two lines were assumed to be non-parallel, we can choose s
and t so that the second term on the right is zero,
and hence d = |(a-b).n| gives the minimal distance.

A picture of all this would be nice too.

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