1. If $|a|=1,|b|=2, \boldsymbol{c}=a+b$, and $c \perp$ $a$, then the angle between vector $a$ and $b$ is degrees.
Given $c \perp a \Rightarrow c \cdot a=0 \Rightarrow(a+b) \cdot a=0$
$$
\Rightarrow|\boldsymbol{a}|^{2}+|\boldsymbol{a}| \cdot|\boldsymbol{b}| \cos \alpha=0 \Rightarrow \cos \alpha=-\frac{1}{2} \text {. }
$$

Therefore, the angle $\alpha$ between vector $a$ and $b$ is $120^{\circ}$.