Find the shortest distance between the lines whose vector equations are ` vec r=( hat i+2 hat j+3 hat k)+lambda( hat i-3 hat j+2 hat k)`and ` vecr=4 hat i+5 hat j+6 hat k+mu(2 hat i+3 hat j+ hat k)`.

Explanation

`vecPQ` = `|[hati, hatj, hatk],[ 1, -3, 2],[ 2, 3,1]|`
= `hati( -3-6) -hatj(1-4) +hatk ( 3+6)`
` = -9hati +3hatj +9hatk`
` = 3( -3hati +hatj +3hatk)`
Shortest distance
`PQ = vec(AB). hat(PQ)`
= `-3( hati +hatj +hatk).(-3hati +hatj +3hatk)/ sqrt( 9+1+9)`
`((-3)(-3+1+3))/sqrt(19)`
= `-3/sqrt(19)`
= `3/sqrt(19)` { taking modulus}