The unit vector which is orthogonal to the vector `5hati + 2hatj + 6hatk ` and is coplanar with vectors `2hati + hatj + hatk and hati - hatj + hatk `is

Options

A. `(2hati - 6hatj + hatk)/sqrt41`
B. `(2hati-3hatj)/sqrt13`
C. `(3 hatj -hatk)/sqrt10`
D. `(4hati + 3hatj - 3hatk)/sqrt34`

Answer

c

Explanation

Any vector coplanar, to `veca and vecb ` can be written as
`vecr=muveca + lamdavecb`
`vecr = (mu + 2lamda) hati + ( -mu+lamda) hatj + ( mu + lamda) hatk`
since `vecr` is othorgonal to `5hati + 2hatj + 6hatk`
` Rightarrow 5 ( mu + 2lambda) +2 ( - mu+ lamda) + 6 ( mu + lambda) = 0`
`or 9mu + 18 lamda =0`
` or lamda =-1/2 mu `
` vecr = lamda ( 3hatj -hatk)`
Since `hatr` is a unit vector s, `hatr = (3hati -hatk) / sqrt10`