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If a,b,c,d are unit vectors such that (axb).(cxd)=1. a.c=1/2 Find the angle between b and d . {please include bar with the vectors}

If a,b,c,d are unit vectors such that (axb).(cxd)=1.
a.c=1/2
Find the angle between b and d .
{please include bar with the vectors}

Grade:12

1 Answers

Sourabh Singh IIT Patna
askIITians Faculty 2104 Points
9 years ago
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 adequation alone if they're parallel, we accept that a x b and c x d are alongside (more accurately co-directional back their dot artefact is positive).

But this agency that a, b ascertain the actual aforementioned even as c, d, back their cantankerous articles are erect to their corresponding plane.

Therefore a, b, c and d are all co-planar assemblage vectors. So 1) and 2) are disqualified out.

Furthermore, back ||a x b|| <= 1 and ||c x d|| <= 1 and ||a x b|| * ||c x d|| = 1, we accept that ||a x b|| = 1 and ||c x d|| = 1 (otherwise we get a atomic contradiction).

This agency that a is erect to b and c is erect to d.

And as apparent above, all four are coplanar assemblage vectors. Accordingly we can visualise them in a assemblage circle.

Now, a.c = 1/2 implies a is at an bend of 60 degrees with c

Since b and d are erect to a and c respectively, this agency the bend amid b and d is either 60 or 120 degrees and accordingly b and d are affirmed not to be parallel.

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