Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes

Question

It's just a doubt.

$\vec { A } = A \hat{ x } + A\hat { y} $

$ A\hat { y} =A\hat{ x} $

$\vec { A } = A \hat { x } + A\hat { x} $

Now finding the magnitud of the vector,

$A=\frac { 1 }{ \sqrt { 2 } } $

So the components of the unit vector are:

$\vec { A } = \frac { 1 }{ \sqrt { 2 } } \hat { x } +\frac { 1 }{ \sqrt { 2 } } \hat { y} $

Is my answer right?