`a.hati=a.(2hati+hatj)=a.(hati+hatj+3hatk)=1,` then `a` is equal to

Options

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

Answer

D

Explanation

Let `a=a_(1)hati+a_(2)hatj+a_(3)hatk`
`because" "a.hati=1`
`implies" "a_(1)=1`
Since, `" "a.(2hati+hatj)=1`
`implies2a_(1)+a_(2)=1impliesa_(2)=1-2`
`impliesa_(2)=-1" and "a.(hati+hatj+3hatk)=1`
`impliesa_(1)+a_(2)+3a_(3)=1implies1-1+3a=1`
`implies" "a_(3)=(1)/(3)`
`:.a=hati-hatj+(1)/(3)hatk=(1)/(3)(3hati-3hatj+hatk)`