The unit vector which is orhtogonal to the vector `3hati+2hatj+6hatk` and is coplanar with vectors `2hati+hatj+hatk` and `hati-hatj+hatk`, is

Options

A. `1/(sqrt(41))(2hati-6hatj+hatk)`
B. `1/(sqrt(13))(2hati-3hatj)`
C. `1/(sqrt(10))(3hatj-hatk)`
D. `1/(sqrt(34))(4hati+3hatj-3hatk)`

Answer

C

Explanation

Let `veca=3hati+2hatj+6hatk, vecb=2hati+hatj+hatk` and `vecc=hati-hatj+hatk`.
Then required unit vectors are given by
`vec(alpha)=+-(veccxx(vecbxxvecc))/(|vecaxx(vecbxxvecc)|)`
Now
`vecaxx(vecbxxvecc)=(veca.vecc)vecb-(veca.vecb)vecc`
`implies(vecaxx(vecbxxvecc)=7(2hati+hatj+hatk)-14(hati-hatj+hatk)`
`impliesvecaxx(vecbxxvecc)=21hatj-7hatk`
`implies |vecaxx(vecbxxvecc)|=sqrt(21^(2)+7^(2))=7sqrt(10)`
Hence required unit vectors are
`vec(alpha)=+-(21hatj-7hatk)/(7sqrt(10))=+-1/(sqrt(10))(3hatj-hatk)`