a
\(\begin{array}{l}
If\,\vec C = a\hat i + b\hat j\,then\,\vec A.\vec C = \vec A.\vec B\\
a + b = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)\\
\vec B.\vec C = \vec A.\vec B\\
2a - b = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)\\
Solving\,equation\,\left( i \right)\,and\,\left( {ii} \right)\,\,we\,get\\
a = \frac{1}{3},\,b = \frac{2}{3}\\
Magnitude\,of\,coplanar\,vector,\\
\left| {\vec C} \right| = \sqrt {\frac{1}{9} + \frac{4}{9}}  = \sqrt {\frac{5}{9}} 
\end{array}\)