Find the shortest distance between the lines`(x+1)/7=(y+1)/(-6)=(z+1)/1`and `(x-3)/1=(y-5)/(-2)=(z-7)/1`

Explanation

The shortest distance between the lines
`(x-x_(1))/(l_(1))=(y-y_(1))/(m_(1))=(z-z_(1))/(n_(1)) ` and `(x-x_(2))/(l_(2))=(y-y_(2))/(m_(2))=(z-z_(2))/(n_(2))` is given by
`d=(|(x_(2)-x_(1),y_(2)-y_(1),z_(2)-z_(1)),(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2))|)/sqrt((m_(1)n_(2)-m_(2)n_(1))^(2)+(l_(2)n_(1)-l_(1)n_(2))^(2)+(l_(1)m_(2)-l_(2)m_(1))^(2))`
Given equations of the lines are
`(x+1)/(7)=(y+1)/(-6)=(z+1)/(1)` and `(x-3)/(1)=(y-5)/(-2)=(z-7)/(1)`
Here `x_(1)=-1, y_(1)=-1,l_(1)=7,m_(1)=-6, n_(1)=1`
Also `x_(2)=3, y_(2)=5,z_(2)=7.l_(2)=1, m_(2)=-2,n_(2)=1`
`:.|(x_(2)-x_(1),y_(2)-y_(1),z_(2)-z_(1)),(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2))|`
`=|(3+1,5+1,7+1),(7,-6,1),(1,-2,1)|`
`=(-6+2)-6(7-1)+8(-14+6)`
`=(-4)-6(6)+8(-8)`
`=-16-36-64=-116`
Also `(m_(1)n_(2)-m_(2)n_(1))^(2)+(l_(2)n_(1)-l_(1)n_(2))^(2)+(l_(1)m_(2)-l_(2)m_(1))^(2)`
`=(-6+2)^(2)+(1-7)^(2)+(-14+6)^(2)`
`=(-6+2)^(2)+(1-7)^(2)+(-14+6)^(2)`
`=(-4)^(2)+(-6)^(2)+(-8)^(2)=116`
`:.` The shortest distance between the given lines
`=|(-116)/(sqrt(116))|=sqrt(116)=2sqrt(29)` units.