Let a plane P contain two lines r → = i ^ + λ i ^ + j ^ , λ ∈ R and r → = - j ^ + μ j ^ - k ^ , μ ∈ R . If Q α , β , γ is the foot of the perpendicular drawn from the point M 1 , 0 , 1 to P , then 3 α + β + γ equals .......

Question

Let a plane P contain two lines r → = i ^ + λ i ^ + j ^ , λ ∈ R and r → = - j ^ + μ j ^ - k ^ , μ ∈ R . If Q α , β , γ is the foot of the perpendicular drawn from the point M 1 , 0 , 1 to P , then 3 α + β + γ equals .......,

MARKS App · Accepted Answer

Normal vector to the plane containing to given lines n → = i ^ j ^ k ^ 1 1 0 0 1 - 1 = - i ^ + j ^ + k ^ Direction ratios of normal vector = - 1 , 1 , 1 . Equation of plane will be - 1 x - 1 + 1 y - 0 + 1 z - 0 = 0 ⇒ x - y - z - 1 = 0 If x , y , z is foot of perpendicular drawn from M 1 , 0 , 1 on the plane, then ⇒ x - 1 1 = y - 0 - 1 = z - 1 - 1 = - 1 - 0 - 1 - 1 3 ⇒ x = 4 3 , y = - 1 3 , z = 2 3 ⇒ α + β + γ = 4 3 - 1 3 + 2 3 = 5 3 ⇒ 3 α + β + γ = 5 .

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