Let `a=2i−j+k` , `b=i+2j−k` and `c=i+j−2k` be three vectors. A vector r in the plane of b and c whose projection on a is of magnitude `sqrt(2/3)` ​ ​ is

Options

A. `2hati+3hatj-3hatk`
B. `2hati+3hatj +3hatk`
C. `-2hati - hatj + 5hatk`
D. `2hati + hatj + 5hatk`

Answer

a,c

Explanation

we have
`veca =2hati-hatj+hatk,vecb=hati+2hatj-hatk,vecc=hati+hatj=2hatk`
Any vector in the plane of `vecb and vecc` is
`vecu=muvecb+lamdavecc`
` mu(hati+2hatj-hatk)+lamda(hati +hatj-2hatk)`
`=(mu+lamda)hati+(2mu+lamda)hatj-(mu + 2lamda)hatk`
Given that the magnitude of projection of `vecu "on " veca` is `sqrt(2//3)` thus
`sqrt(2/3)=|(vecu.veca)/(|veca|)|`
`|(2(mu+lamda)-(2mu+lamda) -(mu+2lamda))/sqrt6|`
`or |-lamda- mu|2`
` Rightarrow lamda + mu=2 or lamda+mu =-2`