In[53]:= Assuming[a [Element] Reals,
Simplify@Normalize[{a,a + 1,a + 2} ]]
Out[53]= {a/Sqrt[5 + 6 a + 3 a^2],(1 + a)/Sqrt[5 + 6 a + 3 a^2],(
2 + a)/Sqrt[5 + 6 a + 3 a^2]}
In[53]:= Assuming[a [Element] Reals,
Simplify@Normalize[{a,a + 1,a + 2} ]]
Out[53]= {a/Sqrt[5 + 6 a + 3 a^2],(1 + a)/Sqrt[5 + 6 a + 3 a^2],(
2 + a)/Sqrt[5 + 6 a + 3 a^2]}