If `overset(to)(a) , overset(to)(b) , overset(to)(c ) " and " overset(to)(d)` are the unit vectors such that
`(overset(to)(a)xx overset(to)(b)). (overset(to)(c )xx overset(to)(d)) =1 " and " overset(to)(a), overset(to)(c ) = .(1)/(2) , ` then

Options

A. `overset(to)(a) , overset(to)(b), overset(to)(c )` are non -coplanar
B. ` overset(to)(a), overset(to)(b), overset(to)(d)` are non- coplanar
C. `overset(to)(b) , overset(to)(d)` are non-parallel
D. `overset(to)(a), overset(to)(d)` are parallel and `overset(to)(b), overset(to)(c )` are parallel

Answer

C

Explanation

Let angle between `vec(a) " and " vec(b) be 0_(1) , vec(c ) and " vec( d) " be " 0_(2) " and " vec(a) xx vec(b)` and `vec(b) xx vec(d)` be 0 .
Since `(vec(a) xx vec(b)). (vec(c ) xx vec(d)) =1`
`rArr sin 0_(1) ." sin "0_(2) ". " cos0 =1`
`rArr 0_(1) =90^(@) , 0_(2) =90^(@) , 0 =0^(@)`
`rArr vec(a) bot vec(b) , vec( c) bot vec(d) , (vec(d) xx vec(d)) ||(vec(c ) xx vec( d))`
`So., vec(a) xx vec(b) = k (vec( c) xx vec(d)) " and " vec(a) xx vec(b) = k(vec(c ) xx vec(d))`
`rArr (vec(a) xx vec(b)) ". " vec(c ) = k (vec( c) xx vec(d)) .(vec(c )`
and `(vec(a) xx vec(b)) ". " vec(d) = k (vec(c ) xx vec(d)) "."vec(d)` ltbgt `rArr (vec(a),vec(b),vec(c )" and " vec(a) ,vec(b), vec(d) ` are coplanar vectors so options (a) and (b) are incorrect.
Let `vec(b)||vec(d) rArr vec(b) =+- vec(d)`
As `(vec(a)xx vec(b)) .(vec(c ) xx vec(d)) =1 rArr (vec(a) xx vec(b)). (vec(c ) xx vec(b)) =+-1`
`rArr [vec(a) xx vec(b) vec(c) vec(b)] =+-1 rArr [vec(c ) vec(b) vec(a) xx vec(b) ]=+-1`
`rArr vec(a ) ." vec(a) =+-1 " "[:' vec(a) "." vec(b) =0]`
Which is a contradiction so
option (c ) is correct.
Let option (d) is correct.

`rArr vec(d) =+- vec(a)`
and `vec( c) = +- vec(b) `
As `(vec( a) xx vec(d))"." (vec(c) xx vec(d)) =1`
`rArr (vec(a) xx vec(d)) .(vec(b)xx vec(a)) =+-1`
Which is a contradiction so option (d) is incorrect,
Alternatively , options (c ) and (d) may be observed from the given figure .