vector a=i+j+k vector b=i-j+k vector c=i-j-k find vector v in the plane of vectors a and b whose projection on vector c is 1/root3

A vector in the plane of b and c is given by

      d = αa + βb

=> d = α(i + j + k) + β(i - j + k)

=> d = i(α + β) + j(α - β) + k(α +β) ...................1

Given length of the projection of d on a = √(2/3)

=> (d . c)/|c| = 1 / √(3)

=> {α + β - α + β -α - β}/{√3}  = 1 / √(3)

=> {β - α } = 1

This satisfies when α = 1 and β = 2

From eqaution 1, we get, d