The coordinates of the foot of the perpendicular drawn from the point A \left(\right. 1 , 0 , 3 \left.\right) to the join of the points B \left(\right. 4 , 7 , 1 \left.\right) and C \left(\right. 3 , 5 , 3 \left.\right) are

Engineering

Mathematics

Line of Greatest Slope

Question

The coordinates of the foot of the perpendicular drawn from the point

A (1, 0, 3)

to the join of the points

B (4, 7, 1)

and

C (3, 5, 3)

are

(5, 7, 17)

(\frac{- 5}{7}, \frac{7}{3}, \frac{- 17}{3},)

(\frac{5}{7}, \frac{- 7}{3}, \frac{17}{3},)

(\frac{5}{7}, \frac{7}{3}, \frac{17}{3},)

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The cartesian equation of the line passing through given 2-point form

\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}}

\frac{x - 3}{1} = \frac{y - 5}{2} = \frac{z - 3}{- 2}

let P be the coordinates of foot

P \equiv (λ + 3, 2 λ + 5, - 2 λ + 3)

DR of

P A \equiv (λ + 2, 2 λ + 5, - 2 λ)

So,

(λ + 2) 1 + (2 λ + 5) 2 + (- 2 λ) (- 2) = 0

9 λ + 12 = 0

λ = - \frac{4}{3}

P \equiv (\frac{5}{3}, \frac{7}{3}, \frac{17}{3})

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