Prove that the plane `vecr*(hati+2hatj-hatk)=3` contains the line `vecr=hati+hatj+lamda(2hati+hatj+4hatk).`

Explanation

To show that `vecr=hati+hatj+lamda(2hati+hatj+4hatk)" "`(i)
lies in the plane `vecr* (hati+2hatj-hatk)=3`,`" "`(ii)
we must show that each point of (i) lies in (ii). In order words we must show that `vecr` in (i) satisfies (ii) for every value of `lamda`. We have
`" "[hati+hatj+lamda(2hati+hatj+4hatk)]*(hati+2hatj-hatk)`
`" "=(hati+hatj)*(hati+2hatj-hatk)+lamda(2hati+hatj+4hatk)*(hati+2hatj-hatk)`
`" "=(1)(1)+(1)(2)+lamda[(2)(1)+(1)(2)+4(-1)]=3+lamda(0)=3`
Hence, line (i) lies in plane (ii).