# Find the coordina

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.

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