Find the shortest distance between the lines gives by
`vecr=(8+3lamda)hati-(9+16lamda)hatj+(10+7lamda)hatk`
and `vecr=15hati+29hatj+5hatk+mu(3hati+8hatj-5hatk)`.

Options

A. `11` units
B. `12` units
C. `13` units
D. `14` units

Answer

D

Explanation

We have `vecr=(8+3lamda)hati-(9+16lamda)hatj+(10+7lamda)hatk`
`=8hati-9hatj+10hatk+3lamdahati-16lamdahatj+7lamdahatk`
`=8hati-9hatj+10hatk+lamda(3hati-16hatj+7hatk)`
`implies veca_(1)=8hati-9hatj+10hatk` and `vecb_(1)=3hati-16hatj+7hatk`
Also `vecr=15hati+29hatj+5hatk+mu(3hati+8hatj-5hatk)`
`implies veca_(2)=15hati+29hatj+5hatk` and `vecb_(2)=3hati+8hatj-5hatk`
Now, shortest distance between two lines is given by `|((vecb_(1)xxvecb_(2)).(veca_(2)-veca_(1)))/(|vecb_(1)xxvecb_(2)|)|`
`:. vecb_(1)xxvecb_(2)=|(hati, hatj, hatk),(3,-16,7),(3,8,-5)|`
`=hati(80-56)-hatj(-15-21)+hatk(24+48)`
`=24hati+36hatj+72hatk`
Now `|vecb_(1)xxvecb_(2)|=sqrt((24)^(2)+(36)^(2)+(72)^(2))`
`=12sqrt(2^(2)+3^(2)+6^(2))=84` ltbgt and `(veca_(2)-veca_(1))=(15-8)hati+(29-9)hatj+(5-10)hatk`
`=7hati+38hatj-5hatk`
`:.` Shortest distance `=|((24hati+36hatj+72hatk).(7hati+38hatj-5hatk))/84|`
`=|(168+1368-360)/84|=|1176/84|=14` units