Find the length and the equations of the line of shortest distance between the lines given by: (x-3)/1=(y-5)/(-2)=(z-7)/1 and (x+1)/7=(y+1)/(-6)=(z+1)/1 The answer given in the book is:2?(29 )units,(x-1)/2=(y-2)/3=(z-3)/4 There is no typing mistake at all.
Asked by Manoj | 23rd May, 2013, 05:48: PM
let P and Q be the points on the given lines, respectively. then the general coordinates of P and Q are:
P(k+3, -2k+5, k+7) and Q (7m-1, -6m-1, m-1)
therefore the direction ratios of PQ are (7m-k-4,-6m+2k-6, m-k-8)
now PQ will be the shortest distance if it is perpendicular to both the given lines, therefore by the condition of perpendicularity,
1(7m-k-4) -2(-6m+2k-6) + 1(m-k-8) = 0 (1)
7(7m-k-4) -6(-6m+2k-6) + 1(m-k-8) = 0 (1)
now solving (1) and (2),
m=0 and k = 0
hence the points are
P(3,5,7) and Q (-1,-1,-1)
therefore the shortest distance between the lines
PQ = sqrt((3+1)^2+(5+1)^2 +(7+1)^2) = sqrt(16+36+64) = sqrt(116) = 2sqrt(29)
since we have the points P and Q, the equation of the line which passes through two given points is:
(x-3)/(3+1) = (y-5)/(5+1) = (z-7)/(7+1)
(x-3)/4 = (y-5)/6 = (z-7)/8
(x-3)/2 = (y-5)/3 = (z-7)/4
However,the answer is given as (x-1)/2=(y-2)/3=(z-3)/4 i.e. the line passes through 1,2,3.
The above equation also passes through 1,2 4
So, they are the same equations.
Answered by | 25th May, 2013, 11:00: PM
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