f the foot of the perpendicular from (1, 2, 3) to the line (x + 1)/2 = (y - 2)/5 = (z - 4)/1 is (α, β, γ), then find α + β + γ
Question
f the foot of the perpendicular from (1, 2, 3) to the line (x + 1)/2 = (y - 2)/5 = (z - 4)/1 is (α, β, γ), then find α + β + γ
Solution
The given line is (x + 1)/2 = (y - 2)/5 = (z - 4)/1. Let's denote the common ratio as λ. So, the general point on the line can be written as P(2λ - 1, 5λ + 2, λ + 4).
The foot of the perpendicular from the point A(1, 2, 3) to the line will be the point on the line such that AP is perpendicular to the line. The direction ratios of the line are [2, 5, 1] and the direction ratios of AP are [2λ - 2, 5λ, λ + 1].
Since AP is perpendicular to the line, the dot product of the direction ratios of AP and the line will be zero. So, we have:
(2λ - 2)2 + 5λ5 + (λ + 1)*1 = 0 4λ - 4 + 25λ + λ + 1 = 0 30λ - 3 = 0 λ = 3/30 = 0.1
Substituting λ = 0.1 in the coordinates of P, we get P(20.1 - 1, 50.1 + 2, 0.1 + 4) = P(-0.8, 2.5, 4.1). So, α = -0.8, β = 2.5, γ = 4.1.
Therefore, α + β + γ = -0.8 + 2.5 + 4.1 = 5.8.
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