What is the angle between the two lines whose direction numbers are `(sqrt3-1,-sqrt3-1,4)and(-sqrt3-1,sqrt3-1,4)`?

Options

A. `pi/6`
B. `pi/4`
C. `pi/3`
D. `pi/2`

Answer

C

Explanation

Let position vectors of two points be
`overset(" "to)(OA)=(sqrt3-1)hati-(sqrt3+1)hatj+4hatk`and
`oversetto(OB)=-(sqrt3+1)hati+(sqrt3+1)hatj+4hatk`
`abs(oversetto(OA))=sqrt((sqrt3-1)^2+(sqrt3+1)^2+4^2)`
`=sqrt(3+1-2sqrt3+3+1+2sqrt3+16)=sqrt24`
Also, `abs(oversetto(OB))=sqrt((sqrt3+1)^2+(sqrt3-1)^2+4^2)=sqrt24`
`oversetto(OA).oversetto(OB)=absoversetto(OA).absoversetto(OB).costheta,"where "theta` is the angle between `oversetto(OA)&oversetto(OB)`.
`oversetto(OA).oversetto(OB)=-(sqrt3-1)(sqrt3+1)-(sqrt3+1)(sqrt3-1)+16`
`=-3+1-3+1+16=12`
So, `costheta(oversetto(OA).oversettoOB)/(absoversetto(OA)absoversetto(OB))=12/24=1/2`
`rArr theta=pi/3`