Question: Two travelers \((P\) and \(Q)\) are moving on lines in space. Their positions at time \(t\) are given by \(P(t)=(5+1,3+2 i, 2+2 t)\) and \(Q(t)=(t, 0, t)\).

1. How close do the travelers ever get?
2. Find the shortest distance between their paths (use a method that enables you to find the points on these two lines that are nearest each other).
3. Give an equation of a plane through \((1,1,1)\) that intersects neither of these lines, or show that it is not possible (the lines are skew lines, meaning they are non-intersecting, non-parallel lines).

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