Ta có:

a) 

$\overrightarrow{a} . \overrightarrow{b}$ = 1 . 3 + 2 . (-4) = -5

$\overrightarrow{b} . \overrightarrow{c}$ = 3 . (-5) + (-4) . 3 = -27

$\overrightarrow{c} . \overrightarrow{a}$ = -5 . 1 + 3 . 2 = 1

b) 

$|\overrightarrow{a}| = \sqrt{1^{2} + 2^{2}} = \sqrt{5}$

$\overrightarrow{b} + \overrightarrow{c} = (-2; -1)$

$|\overrightarrow{b} + \overrightarrow{c}| = \sqrt{(-2)^{2} + (-1)^{2}} = \sqrt{5}$

$\overrightarrow{a} . (\overrightarrow{b} + \overrightarrow{c})$ = 1 . (-2) + 2 . (-1) = -4

$cos(\overrightarrow{a}; \overrightarrow{b} + \overrightarrow{c}) = \frac{\overrightarrow{a} . (\overrightarrow{b} + \overrightarrow{c})}{|\overrightarrow{a}| . |\overrightarrow{b} + \overrightarrow{c}|} = \frac{-4}{\sqrt{5} . \sqrt{5}} = \frac{-4}{5}$

Vậy $(\overrightarrow{a}; \overrightarrow{b} + \overrightarrow{c}) \approx 143^{o}$