A unit vector coplanar with `veci + vecj + 2veck and veci + 2 vecj + veck` and perpendicular to `veci + vecj + veck ` is _______

Answer

`(hatj -hatk)/(sqrt2)or(-hatj+hatk)/(sqrt2)`

Explanation

Let `xhati + yhatj +zhatk` be a unit vector coplanar with
`hati + hatj + 2hatk and hati + 2hatj +hatk` and also perpendicular to `hati + hatj + hatk` . Then
`|{:(x,y,z),(1,1,2),(1,2,1):}|=0`
`or -3x+ y+z=0`
` and x+y+z=0`
Solving the above by cross - product method , we get
`x/0=y/4=z/(-4)orx/0=y/1=z/(-1)=lambda(say)`
`Rightarrow x=0,y=lamda,z=-lambda`
As `xhati+yhatj+zhatk` is a unit vector , we have
` 0 + lambda^(2) +lambda^(2) =1 `
` or lamda^(2) = 1/2 or lambda= +- 1/sqrt2`
Required vector = `(hatj- hatk)/ sqrt2 or (-hatj +hatk)/sqrt2`