Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line Also, find the length of the perpendicular from the given point to the line.

Given: Equation of line is

To find: coordinates of foot of the perpendicular from (1, 2, 3) 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 (3λ + 6, 2λ + 7, -2λ + 7)

Direction ratio of the line is 3 : 2 : -2

Direction ratio of the perpendicular is

⇒ (3λ + 6 - 1) : (2λ + 7 - 2) : (-2λ + 7 - 3)

⇒ (3λ + 5) : (2λ + 5) : (-2λ + 4)

Since this is perpendicular to the line,

3(3λ + 5) + 2(2λ + 5) – 2(-2λ + 4) = 0

⇒ 9λ + 15 + 4λ + 10 + 4λ – 8 = 0

⇒ 17λ = -17

⇒ λ = -1

So the foot of the perpendicular is (3, 5, 9)

Distance

= 7 units

Therefore, the foot of the perpendicular is (3, 5, 9) and length of perpendicular is 7 units.