The shortest distance between the lines
` r = ( - hati - hatj - hatk ) + lamda ( 7 hati - 6 hatj + hatk ) and r = ( 3 hati + 5 hatj + 7 hatk ) + mu ( hati - 2 hatj + hatk ) `

Options

A. ` sqrt ( 29 ) ` units
B. 29 units
C. ` ( 2 9 ) / ( 2 ) ` units
D. ` 2 sqrt ( 29 ) ` units

Answer

D

Explanation

The given lines are
` r = - hati - hatj - hatk+ lamda ( 7 hati - 6hatj + hatk ) ` and
` r = ( 3 hati + 5 hatj + 7 hatk ) + mu ( hati - 2hatj + hatk ) `
where, ` a _ 1 = - hati - hatj - hatk , b _ 1 = 7 hati - 6 hatj + hatk `
and ` a_ 2 = 3hati + 5 hatj + 7 hatk , b _ 2 = hati - 2 hatj + hatk `
Now, ` a _ 2- a _ 1 = ( 3 hati + 5hatj + 7 hatk ) - ( - hati - hatj - hatk ) `
` = 4 hati + 6hatj + 8 hatk `
and ` b _ 1 xx b _ 2 = |{:( hati , hatj , hatk ) , ( 7, - 6, 1 ) , ( 1, -2, 1 ) :}| `
` = hati ( - 6 + 2 ) - hatj ( 7 - 1 ) + hatk ( -14 + 6 ) `
` = - 4hati - 6 hatj - 8 hatk `
` rArr | b _ 1 xx b _ 2 | = sqrt(( - 4 ) ^ 2 + ( - 6 ) ^ 2 + ( -8 ) ^ 2 ) `
` = sqrt ( 16 + 36 + 64 ) = sqrt ( 116 ) `
` therefore ` Shortest distance between the given lines
` d= | (( b _ 1 xx b _ 2 ) * ( a _ 2 - a _ 1 ) ) / ( |b _ 1 xx b _ 2 | ) | `
` (| ( - 4hati - 6 hatj - 8 hatk ) * ( 4hati + 6 hatj + 8 hatk )| ) / ( sqrt ( 116 ) ) `
` = (| ( -4 ) xx 4 + ( - 6 ) xx 6 + ( - 8 ) xx 8 | ) /( sqrt ( 116 )) `
` = (116 ) / ( sqrt (116 )) = sqrt (116) = 2 sqrt (29 ) ` units