If `veca=hati+hatj+hatk and vecb=hatj-hatk` find a vector `vecc` such that `vecaxxvecc=vecb` and `veca.vecc=3`.

Explanation

Let `vecc=xhati+yhatj+xhatk`
Also, `veca=hati+hatj+hatkandvecb=hatj-hatk`
For `vecaxxvecc=vecb`. `|[hati,hatj,hatk],[1,1,1],[x,y,z]|=hatj-hatk`
`implieshatj(z-y)-hatj(z-x)+hatk(y-x)=hatj-hatk`
`:.z-y=0......(i)`
`x-z=1.......(ii)`
`x-y=1......(iii)`
Also, `veca.vecc=3`
`(hati+hatj+hatk).(xhati+yhatj+xhatk)=3`
`impliesx+y+z=3......(iv)`
On adding Eqs. (ii) and (iii). we get
`2k-y-z=2......(v)`
On solving Eqs. (iv) and (v), we get
`x(5)/(3)`
`:.y=(5)/(3)-1=(2)/(3)andz=(2)/(3)`
Now, `vecc=(5)/(3)hati+(2)/(3)hatj+(2)/(3)hatk`
`=(1)/(3)(5hati+2hatj+2hatk)`