a) Ta có: \(2\overrightarrow b = \left( {2;2; - 2} \right)\).

Do đó, \(\overrightarrow a - 2\overrightarrow b = \left( { - 2 - 2;1 - 2;2 - \left( { - 2} \right)} \right) = \left( { - 4; - 1;4} \right)\).

Vậy \(\overrightarrow u = \left( { - 4; - 1;4} \right)\).

b) Ta có: \(\left| {\overrightarrow u } \right| = \sqrt {{{\left( { - 4} \right)}^2} + {{\left( { - 1} \right)}^2} + {4^2}} = \sqrt {33} \).

Vậy độ dài vectơ \(\overrightarrow u \) là \(\sqrt {33} \).

c) Ta có: \(\cos \left( {\overrightarrow a ,\overrightarrow b } \right) = \frac{{\overrightarrow a .\overrightarrow b }}{{\left| {\overrightarrow a } \right|.\left| {\overrightarrow b } \right|}} = \frac{{ - 2.1 + 1.1 + 2.\left( { - 1} \right)}}{{\sqrt {{{\left( { - 2} \right)}^2} + {1^2} + {2^2}} .\sqrt {{1^2} + {1^2} + {{\left( { - 1} \right)}^2}} }} = \frac{{ - \sqrt 3 }}{3}\).

Vậy \(\cos \left( {\overrightarrow a ,\overrightarrow b } \right) = \frac{{ - \sqrt 3 }}{3}\).