Q. 3

Find the length and the equations of the line of shortest distance between the lines given by:

and

Answer :

Given : Cartesian equations of lines




Formulae :


1. Condition for perpendicularity :


If line L1 has direction ratios (a1, a2, a3) and that of line L2 are (b1, b2, b3) then lines L1 and L2 will be perpendicular to each other if



2. Distance formula :


Distance between two points A≡(a1, a2, a3) and B≡(b1, b2, b3) is given by,



3. Equation of line :


Equation of line passing through points A≡(x1, y1, z1) and B≡(x2, y2, z2) is given by,



Answer :


Given equations of lines




Direction ratios of L1 and L2 are (2, 1, -3) and (2, -7, 5) respectively.


Let, general point on line L1 is P≡(x1, y1, z1)


x1 = 2s-1 , y1 = s+1 , z1 = -3s+9


and let, general point on line L2 is Q≡(x2, y2, z2)


x2 = 2t+3 , y2 = -7t – 15 , z2 = 5t + 9





Direction ratios of are ((5t - 2s + 10), (-7t – s – 16), (5t + 3s))


PQ will be the shortest distance if it perpendicular to both the given lines


Therefore, by the condition of perpendicularity,


2(5t - 2s + 10) + 1(-7t – s – 16) - 3(5t + 3s) = 0 and


2(5t - 2s + 10) – 7(-7t – s – 16) + 5(5t + 3s) = 0


10t – 4s + 20 - 7t – s - 16 - 15t – 9s = 0 and


10t - 4s + 20 + 49t + 7s + 112 + 25t + 15s = 0


-12t – 14s = -4 and


84t + 18s = -132


Solving above two equations, we get,


t = -2 and s = 2


therefore,


P ≡ (3, 3, 3) and Q ≡ (-1, -1, -1)


Now, distance between points P and Q is







Therefore, the shortest distance between two given lines is



Now, equation of line passing through points P and Q is,







Therefore, equation of line of shortest distance between two given lines is


x = y = z


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