If a,b,c and d are unit vectors such that (axb).(cxd)=1 and a.c=1/2 then
A] a,b,c are non-coplanar
B]b,c,d are non co-planar
C]b,d are non parallel
D]a,d are parallel and b,c are parallel
plsgive explaination too for ur choice
Given ||a x b|| <= ||a|| ||b|| = 1
and ||c x d|| <= ||c|| ||d|| = 1
and that | (a x b) . (c x d) | <= ||a x b|| * ||c x d|| = 1, with equality only when they're parallel, we have that a x b and c x d are parallel (more specifically co-directional since their dot product is positive).
But this means that a, b define the very same plane as c, d, since their cross products are perpendicular to their respective plane.
Therefore a, b, c and d are all co-planar unit vectors. So 1) and 2) are ruled out.
Furthermore, since ||a x b|| <= 1 and ||c x d|| <= 1 and ||a x b|| * ||c x d|| = 1, we have that ||a x b|| = 1 and ||c x d|| = 1 (otherwise we get a trivial contradiction).
This means that a is perpendicular to b and c is perpendicular to d.
And as shown above, all four are coplanar unit vectors. Therefore we can visualise them in a unit circle.
Now, a.c = 1/2 implies a is at an angle of 60 degrees with c
Since b and d are perpendicular to a and c respectively, this means the angle between b and d is either 60 or 120 degrees and therefore b and d are guaranteed not to be parallel.
Therefore 3) is correct