Question

Let PM be the perpendicular from the point P(1, 2, 3) to xy plane. If vector OP makes an angle θ with the positive direction of z-axis and vector OM makes an angle ϕ with the positive direction of x-axis, where O is the origin and θ and ϕ are acute angles, then 

(A) tanθ = √5/3 

(B) sinθsinϕ = 2/√14

(C) tanϕ = 2 

(D) cosθcosϕ = 1/√14

Answer 1

Answer is (A), (B), (C)

See Fig. 

If P be (x, y, z) then from the figure, x = r sinθ cosϕ, y = r sinθ sinϕ, z = r cosθ

⇒ 1 = r sinθ cosϕ, 2 = r sinθ sinϕ, 3 = r cosθ

⇒ 1² + 2² + 3² = r² ⇒ r = ± √14

Therefore,

sinθ cosϕ = 1/√14 sinθsinϕ = 2/√14, cosϕ = 3/√14

(neglecting negative sign as q and f are acute) 

Therefore,