#### Question

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

#### Solution

We know that the shortest distance between the lines

`bar r=bara_1+lambdabarb_1 and barr=bara_2+mubarb_2 `

Given equation of lines are

`bar r=(4hati-hatj)+lambda(hati+2hatj-3hatk) and barr=(hati-hatj+2hatk)+mu(hati+4hatj-5hatk)`

`therefore bara_2-bara_1=(hati-hatj+2hatk)-(4hati-hatj)=-3hati+2hatk`

`barb_1xxbarb_2=|[hati,hatj,hatk],[1,2,-3],[1,4,-5]|`

`=hati(-10+12)-hatj(-5+3)+hatk(4-2)`

`=2hati+2hatj+2hatk`

`(bara_2-bara_1).(barb_1xxbarb_2)=(-3hati+2hatk).(2hati+2hatj+2hatk)=-3xx2+0xx2+2xx2=-2`

`|barb_1xxbarb_2|=sqrt(4+4+4)=2sqrt3`

Shortest distance =

`d=|-2/(2sqrt3)|=|-1/sqrt3|=1/sqrt3 units.`