Let
θ1 be the angle between a and b and
θ2 be the angle between c and d
Now, a × b = |a||b| sin
θ1 n1 =
sinθ1n1 and c × b = |c| |d|
sinθ2n2 =
sinθ2n2
where
n1 is a unit vector parallelogram to the plane of a and b; and
n2 is a unit vectorperpendicular to the plane of c and d
As (a × b) . (c × d) = 1 , we get (
sinθ1) (
sinθ2)
n1.n2 = 1
(
sinθ1) (
sinθ2)
|n1||n2| cosϕ = 1 where
Ï• is angle between
n1 and
n2 ⇒ (
sinθ1) (
sinθ2) =
cosϕ = 1 ⇒
θ1 = π/2 ,
θ2 = π/2 and
Ï• = 0
As
Ï• = 0 we get
n1 and
n2are parrel
Therefore, a, b, c and d are coplanar
also a.c = 1/2 ⇒ |a| |c| cosθ = 1/2
So that angle between a and c is π/3
⇒ the angle between b and d is π/3
∴ b and d are non planar