Since the tetrahedron is regular,

So, OA = OB = OC = AB = BC = CA

Now, AB = BC = CA

=> AB² = BC² = CA²

=> |b - a|² = |c - b|² = |a - c|²     {Here a, b, c are position vectors for A, B and C are respectively}

=> |b|² + |a|² - 2a.b = |c|² + |b|² - 2b.c = |a|² + |c|² - 2a.c

=> a.b = b.c = c.a

=> a.b = b.c

=> a.b - b.c = 0

=> b.(a - c) = 0

=> OB. AC = 0

=> OB is perpendicular to AC

So, any two opposite edges in a regular tetrahedron are perpendicular.