The vector(s) which is/are coplanar with vectors ` hat i+ hat j+2 hat k and hat i+2 hat j+ hat k ,` and perpendicular to vector ` hat i+ hat j+ hat k ,` is/are a.` hat j- hat k` b. `- hat i+ hat j` c. ` hat i- hat j` d. `- hat j+ hat k`

Options

A. `hatj - hatk`
B. `-hati + hatj`
C. `hati -hatj`
D. `-hatj +hatk`

Answer

a,d

Explanation

Any vector in the plane of `veca = hati +hatj + 2hatk and vecb = hati + 2hatj + hatk` is
`vecr=lamda(hati+hatj+2hatk) +mu(hati+2hatj+hatk)`
`= (lamda+mu)hati+(lamda+2mu)hatj= (2lamda+mu)hatk`
Also `vecr` is perpendicular to the vector `hati + hatj +hatk`
`Rightarrow vecr.vecc=0`
` Rightarrow lamda + mu =0`
Possible vectors are `hatj -hatk or -hatj +hatk`