# Find the length a

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.

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