Given: – Perpendicular from A(1, 8, 4) drawn at line joining points B(0, – 1, 3) and C(2, – 3, – 1)

and D be the foot of the perpendicular drawn from A(1, 8, 4) to line joining points B(0, – 1, 3) and C(2, – 3, – 1).

Now let's find the equation of the line which is formed by joining points B(0, – 1, 3) and C(2, – 3, – 1)

Tip: - Equation of a line joined by two points A(x₁,y₁,z₁) and B(x₂,y₂,z₂) is given by

Now

Therefore,

⇒ x = 2λ, y = – 2λ – 1, z = – 4λ + 3

Therefore, coordinates of D(2λ, – 2λ – 1, – 4λ + 3)

Now as we know (TIP) ‘if two points A(x₁,y₁,z₁) and B(x₂,y₂,z₂) on a line, then its direction ratios are proportional to (x₂ – x₁,y₂ – y₁,z₂ – z₁)’

Hence

Direction Ratios of AD

= (2λ – 1), ( – 2λ – 1 – 8), ( – 4λ + 3 – 4)

= (2λ – 1), ( – 2λ – 9), ( – 4λ – 1)

and by comparing with given line equation, direction ratios of the given line are

(hint: denominator terms of line equation)

= (2, – 2, – 4)

Since AD is perpendicular to given line, therefore by “condition of perpendicularity”

a₁a₂ + b₁b₂ + c₁c₂ = 0 ; where a terms and b terms are direction ratio of lines which are perpendicular to each other.

⇒ 2(2λ – 1) + ( – 2)( – 2λ – 9) – 4( – 4λ – 1) = 0

⇒ 4λ – 2 + 4λ + 18 + 16λ + 4 = 0

⇒ 24λ + 20 = 0

⇒

Therefore coordinates of D

i.e Foot of perpendicular

By putting value of λ in D coordinate equation, we get