Given: Equation of line is

To find: coordinates of foot of the perpendicular from (2, -1, 5) to the line. And find the length of the perpendicular.

Formula Used:

1. Equation of a line is

Cartesian form:

where is a point on the line and b₁ : b₂ : b₃ is the direction ratios of the line.

2. Distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is

Explanation:

Let

So the foot of the perpendicular is (10λ + 11, -4λ - 2, -11λ - 8)

Direction ratio of the line is 10 : -4 : -11

Direction ratio of the perpendicular is

⇒ (10λ + 11 - 2) : (-4λ - 2 + 1) : (-11λ - 8 - 5)

⇒ (10λ + 9) : (-4λ - 1) : (-11λ - 13)

Since this is perpendicular to the line,

10(10λ + 9) - 4(-4λ - 1) - 11(-11λ - 13) = 0

⇒ 100λ + 90 + 16λ + 4 + 121λ + 143 = 0

⇒ 237λ = -237

⇒ λ = -1

So the foot of the perpendicular is (1, 2, 3)

Distance

= √14 units

Therefore, the foot of the perpendicular is (1, 2, 3) and length of perpendicular is √14 units.