1.1.45 Vector Algebra

Condition: in the rectangular basis \ (b = (i, j, k) \) vector \ (a \) has a decomposition \ (a = -2 i+j-k \). Make sure that the triple of vectors \ [i^{\ prime} = i, j^{\ prime} = \ frac {1} {\ sqrt {2} j- \ frac {1} {2}} k, \ quad k^{\ prime} = \ frac {1} {\ sqrt {2}} j+\ frac {1} {\ sqrt {2} k \] also forms a rectangular basis \ ( B^{\ Prime} = \ Left (I^{\ Prime}, J^{\ Prime}, K^{\ Prime} \ Right) \), and find the coordinate of the vector \ (A \) in this basis.