$${\begin {array} {l} \vec a \circ (\vec b + \vec c) = a_1 \circ (b_1 + c_1) + a_2 \circ (b_2 + c_2) \\ = (a_1 \cdot b_1 + a_1 \cdot c_1) + (a_2 \cdot b_2 + a_2 \cdot c_2) \\ = \vec a \circ \vec b + \vec a \circ \vec c \end {array}}$$