The vectors which is/are coplanar with vectors `hati+hatj+2hatk and hati+2hatj+hatk` and perpendicular to the vector `hati+hatj+hatk ` is /are (A) `hatj-hatk` (B) `-hati+hatj` (C) `hati-hatj` (D) `-hatj+hatk`

Options

A. `((hatj-hatk)/(sqrt(2)))`
B. `((hati+hatj+hatk)/(sqrt(3)))`
C. `((hati+hatj+2hatk)/(sqrt(6)))`
D. `((hati+2hatj+hatk)/(sqrt(6)))`

Answer

A

Explanation

Let unit vector be `a hai+b haj+c hak`
which is perpendicular to `hati+hatj+hatk`
`thereforea+b+c=0` . . . (i)
and `ahai+bhatj+chak,hai+hatj+2hatk and hati+2hatj+hatk` are coplanar.
`therefore|(a,b,c),(1,1,2),(1,2,1)|=0`
`implies-3a+b+c=0` . . . (ii)
On solving eqs. (i) and (ii) we get
a=0 and c=-b
Also, `ahati+bhatj+chatk` is a unit vector, then
`a^(2)+b^(2)+c^(2)=1`
`implies0+b^(2)+b^(2)=1impliesb=(1)/(sqrt(2))`
`thereforeahati+bhatj+chatk=(1)/(sqrt(2))hatj-(1)/(sqrt(2))hatk`
`=(hatj-hatk)/(sqrt(2))`