Distributivgesetz beim Skalarprodukt
The geometric definition gives us the dot product as the magnitude of $\vec{a}$ multiplied by the scalar projection of $\vec{b}$ onto $\vec{a}$. This is given for any $\vec{a}$, $\vec{b}$ in n-space.
$$\vec{a}\cdot\vec{b}=|\vec{a}|\cdot|\vec{b}|\cdot cos(\vartheta)=|\vec{a}|\cdot|\vec{b}_{\vec{a}}|$$
The dot product of $\vec{a}$ with $\vec{b}+\vec{c}$ is just the magnitude of $\vec{a}$ times the scalar projection of $\vec{b}+\vec{c}$ onto $\vec{a}$. But that can be broken up into components, after which normal distribution takes over.
$$\begin{aligned}\vec{a}\cdot\left(\vec{b}+\vec{c}\right)&=|\vec{a}|\cdot\left|\left(\vec{b}+\vec{c}\right)_{\vec{a}}\right|\\&=|\vec{a}|\cdot\left(|\vec{b}_{\vec{a}}|+|\vec{c}_{\vec{a}}|\right)\\&=|\vec{a}|\cdot|\vec{b}_{\vec{a}}|+|\vec{a}|\cdot|\vec{c}_{\vec{a}}|\\&=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}\end{aligned}$$