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 ) = ( 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 _ 1m _ 2 - l_ 2 m _ 1 ) ^(2)) `
The equations of the given lines are
` ( x + 1 ) / ( 7 ) = ( y + 1 ) / ( - 6 ) = ( z + 1 ) / (1 ) and ( x- 3 )/ (1 )= ( y - 5 ) / ( - 2 ) = ( z - 7 )/ ( 1 )`
` therefore x_ 1 = -1, y_ 1 = - 1, z _ 1 = - 1, x _2 = 3, y _ 2 = 5, z _ 2 = 7`,
` l _ 1 = 7, m _ 1 = - 6, n _ 1 = 1, l _ 2 = 1, m _ 2 = - 2 , n _ 2 = 1 `
` therefore |{:( x _ 2- x _ 1,, y _ 2 - y _ 1,, z _ 2 - z _ 1 ) , ( l _ 1 ,, m _ 1 ,,n _ 1 ) , ( l _ 2,, m_2,, n _ 2) :}| = |{:( 4,, 6,, 8), ( 7, , - 6,, 1 ), ( 1,, - 2,, 1) :}| `
` = 4 ( - 6 + 2 ) - 6 ( 7 - 1) + 8 ( - 14 + 6 ) `
` = - 16 - 36 - 64 = - 116 `
and ` ( m _ 1 n _ 2 - m _ 2n _ 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 `
` = 16 + 36 + 64 = 116 `
Hence, the required shortest distance between the given lines
` = | ( - 116)/ ( sqrt ( 116))| = sqrt ( 116) = 2 sqrt ( 29) ` units.