Find the shortest distance between the lines `bar(r) = (4hati - hatj) + lamda(hati + 2hatj - 3hatk)` and ` bar(r) = (hati - hatj + 2hatk) + mu(hati + 4hatj - 5hatk).`

Explanation

we know that the shortest distance between the skew lines `bar(r) = bar(a_(1)) + lamda bar(b_(1))` and `bar(a_(2))+ mubar(b_(2))` is given by
`d = |((barb_(1)xxbarb_(2)). (bara_(2) - bar(a_(1))))/(|barb_(1)xxbarb_(2)|)|`.
Here` bar a_(1) = 4 hati - hatj + 2 hatk`,
` bar ( b _ 1 ) = hati + 2 hatj - 3 hatk, bar ( b _ 2 ) = hati + 4 hatj - 5hatk `.
` therefore bar ( b _ 1 ) xx bar ( b _ 2 ) = |{:( hati ,, hatj ,,hatk ) , ( 1,,2 ,, - 3 ) , ( 1,,4,,- 5 ) :}| `
= ` ( - 10 + 12 ) hati - ( - 5 + 3 ) hatj + ( 4-2 ) hatk ) `
` = 2 hati + 2 hatj + 2 hatk `
and ` bar ( a _ 2 ) - bar ( a _ 1 ) = ( hati - hatj + 2 hatk ) - ( 4 hati - hatj) `
= ` - 3 hati + 2 hatk `
` therefore ( bar ( b_ 1 ) xx bar ( b _2)) * ( bar ( a _ 2 ) - bar ( a _ 1) ) = ( 2hati + 2hatj + 2 hatk )* ( -3hati + 2 hatk ) `
` = 2 ( - 3 ) + 2 ( 0 ) + 2 (2 ) `
` = - 6 + 0 + 4 = - 2 `
and ` | bar ( b _ 1 ) xx bar ( b _ 2 ) | = sqrt( 2 ^ 2 + 2 ^2 2 ^2) `
` = sqrt ( 4 + 4 + 4 ) = 2sqrt( 3) `
` therefore ` required shortest distance between the given lines ` = | ( - 2 ) /( 2 sqrt 3 ) | = ( 1 ) /(sqrt3 ) ` units.