1.1.45 Vector Algebra

condition: In a rectangular basis \( B=(i, j, k) \) the vector \( a \) has a decomposition \( a=-2 i+j-k \). Make sure that the triple of vectors \[ i^{\prime}=i, j^{\prime}=\frac{1}{\sqrt{2}} j-\frac{1}{\sqrt{2}} k, \quad k^{\prime}=\frac{1}{\sqrt{2}} j+\frac{1}{\sqrt{2}} k \] also forms a rectangular basis \( B^{\prime}=\left(i^{\prime}, j^{\prime}, k^{\prime}\right) \), and find the coordinates of the vector \( a \) in this basis.