JEE Mains Model Paper 25

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 ϕ is angle between n1 and n2
⇒ (sinθ1) (sinθ2) = cosϕ = 1 ⇒ θ1 = π/2 , θ2 = π/2 and ϕ = 0
As ϕ = 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