Let `L_(1)` and `L_(2)` denote the lines `vecr=veci+lamda(-hati+2hatj+2hatk),lamdainR` and `vecr=mu(2hati-hatj+2hatk),muinR`
Respectively if `L_(3)` is a line which is perpendicular to both `L_(1)` and `L_(2)` and cuts both of them, then which of the following options describe(s) `L_(3)`?

Options

A. `vecr=t(2hati+2hatj-hatk),tinR`
B. `vecr=(2)/(9)(4hati+hatj+hatk),t(2hati+2hatj-hatj),tin R`
C. `vecr=(1)/(3)(2hati+hatj)+t(2hati+2hatj-hatk),t in R`
D. `vecr=(2)/(9)(2hati-hatj+2hatk)+t(2hati+2hatj-hatk),t in R`

Answer

B, C, D

Explanation

Both given lines are skew lines
so direction ratios of any line perpendicular to these lines are `6hati+6hatj-3hatk`
`lt 2, 2 ,-1 gt`
Points at shortest distance between given lines are
`vec(AB)bot"line"L_(1)`
`vec(AB)bot"line"L_(2)`
So `A((8)/(9),(2)/(9),(2)/(9))`
Now equation of required tie `vecr=((8)/(9)hati+(2)/(9)hatj+(2)/(9)hatk)+alpha(2hati+2hatj-hatk)`
Now by option B, C, D are correct.