Answer :

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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