What is linear extension g of f such that $||f||=||g||$?

Question

Consider a functional $f$ on the Euclidean plane $R^2$ defined by $f(x) = ax_1+bx_2$ where $x = (x_1,x_2)$. What is its linear extension $g$ to $R^3$ such that $||f||=||g||$ ?

I took $g(x_1,x_2,x_3)= ax_1+bx_2+cx_3$ but here $||f||\ne||g||$