If `vecA=(1,1,1) and vecC=(0,1,-1)` are given vectors then find a vector `vecB` satisfying equations `vecAxxvecB=vecC and vecA.vecB=3`

Answer

`5/3hati+2/3hatj + 2/3 hatk`

Explanation

Given `vecA = hati + hatj - hatk and vecC = hatj -hatk`
Let `vecB =x hati + yhatj +zhatk`
Given that `vecAxxvecB=vecC Rightarrow |{:(hati,hatj,hatk),(1,1,1),(x,y,z):}|=hatj=hatk`
`or (z-y)i +(x-z)hatj +(y-x)hatk=hatj-hatk`
z-y=0 , x-z=1 and y-x =-1
Also , `vecA. vecB=3`
`Rightarrow x+y +z=3`
Using (i) and(ii) , we get
`y=2//3,xx=5//3,z=2//3`
`vecB = 5/3 hati + 2/3hatj + 2/3 hatk`