Let a unit vector O P → make angle α , β , γ with the positive directions of the co-ordinate axes OX , OY , OZ respectively, where β ∈ 0 , π 2 . O P → 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 O P → make angle α , β , γ with the positive directions of the co-ordinate axes OX , OY , OZ respectively, where β ∈ 0 , π 2 . O P → 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 ∈ π 2 , π and γ ∈ π 2 , π, α ∈ 0 , π 2 and γ ∈ 0 , π 2, α ∈ π 2 , π and γ ∈ 0 , π 2, α ∈ 0 , π 2 and γ ∈ π 2 , π

MARKS App · Accepted Answer

Equation of plane passing through the points 1 , 2 , 3 , 2 , 3 , 4 and 1 , 5 , 7 is 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 Direction ratios of normal of the plane is 1 , - 4 , 3 > Direction cosines of normal of the plane is " open=" ± 1 26 , ∓ 4 26 , ± 3 26 Hence, cos β = 4 26 > 0 ⇒ β ∈ 0 , π 2 , cos α = - 1 26 ⇒ α ∈ π 2 , π , cos γ = - 3 26 ⇒ γ ∈ π 2 , π

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