Question:
(I)- ,if two pairs opposite edges of a tetrahedron are perpendicular , then prove that opposite edges of third pair are also perpendicular to each other .
Answer:
Let ABCD is a tetrahedron and AD, BC and AC, BD are two pairs of opposite edges which are perpendicular to each other.
i.e. AD.BC = 0 and AC.BD = 0
Let OA = a, OB = b, OC = c and OD = d
Now, AD.BC = 0
=> (d - a).(c - b) = 0 ..............1
and AC.BD = 0
=> (c - a).(d - b) = 0 .............2
Now, from equation 1 - equation 2, we get
(d - a).(c - b) - (c - a).(d - b) = 0
=> (d.c - d.b - a.c + a.b) - (c.d - c.b - a.d + a.b) = 0
=> d.c - d.b - a.c + a.b - c.d + c.b + a.d - a.b = 0
=> c.b - d.b + a.d - a.c = 0
=> (c - d).b + a.(d - c) = 0
=> (c - d).b - a.(c - d) = 0
=> (c - d).(b - a) = 0
=> CD.AB = 0
Hence, the third pair of opposite edges AB, CD are also perpendicular to each other.
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