Let a unit vector widehat OP make angles α, β, γ with the positive directions of the co-ordinate axes OX, OY , OZ respectively, where β ∈(0, (π/2)). If widehat OP is perpendicular to the plane through points (1,2, 3),(2,3,4) and (1,5,7), then which one of the following is true ?

Question

Let a unit vector widehat OP make angles α, β, &gamma; with the positive directions of the co-ordinate axes OX, OY , OZ respectively, where β ∈(0, (π/2)). If widehat OP is perpendicular to the plane through points (1,2, 3),(2,3,4) and (1,5,7), then which one of the following is true ? (A) α ∈(0, (π/2)) and &gamma; ∈(0, (π/2)) (B) α ∈((π/2), π) and &gamma; ∈((π/2), π) (C) α ∈((π/2), π) and &gamma; ∈(0, (π/2)) (D) α ∈(0, (π/2)) annd &gamma; ∈((π/2), π)

Tardigrade Admin · Accepted Answer

Equation of plane :- | x-1 y-2 z-3 1 1 1 0 3 4 |=0 ⇒[x-1]-4[y-2]+3[z-3]=0 ⇒ x-4 y+3 z=2 D.R's of normal of plane <1,-4,3> D.C's of langle± (1/√26), ∓ (4/√26), ± (3/√26) rangle cos β=(4/√26) cos α=(-1/√26) (π/2) < α < π cos &gamma;=(-3/√26) (π/2) < &gamma; < π

$α \in (0, \frac{π}{2})$ and $γ \in (0, \frac{π}{2})$

$α \in (\frac{π}{2}, π)$ and $γ \in (\frac{π}{2}, π)$

$α \in (\frac{π}{2}, π)$ and $γ \in (0, \frac{π}{2})$

$α \in (0, \frac{π}{2})$ annd $γ \in (\frac{π}{2}, π)$

Q. Let a unit vector OP make angles α,β,γ with the positive directions of the co-ordinate axes OX, OY,OZ respectively, where β∈(0,2π​). If OP is perpendicular to the plane through points (1,2, 3),(2,3,4) and (1,5,7), then which one of the following is true ?

α∈(0,2π​) and γ∈(0,2π​)

α∈(2π​,π) and γ∈(2π​,π)

α∈(2π​,π) and γ∈(0,2π​)

α∈(0,2π​) annd γ∈(2π​,π)

Equation of plane :- ∣∣​x−110​y−213​z−314​∣∣​=0 ⇒[x−1]−4[y−2]+3[z−3]=0 ⇒x−4y+3z=2 D.R's of normal of plane <1,−4,3> D.C's of ⟨±26​1​,∓26​4​,±26​3​⟩ cosβ=26​4​ cosα=26​−1​2π​<α<π cosγ=26​−3​2π​<γ<π

$α \in (0, \frac{π}{2})$ and $γ \in (0, \frac{π}{2})$

$α \in (\frac{π}{2}, π)$ and $γ \in (\frac{π}{2}, π)$

$α \in (\frac{π}{2}, π)$ and $γ \in (0, \frac{π}{2})$

$α \in (0, \frac{π}{2})$ annd $γ \in (\frac{π}{2}, π)$