[Math] Proving that the dot product is distributive

Question

I know that one can prove that the dot product, as defined "algebraically", is distributive. However, to show the algebraic formula for the dot product, one needs to use the distributive property in the geometric definition. How would one show, geometrically, that for Euclidean vectors $\mathbf{a},\mathbf{b},\mathbf{c}$, $$\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\cdot\mathbf{c}=\mathbf{a}\cdot(\mathbf{b}+\mathbf{c})?$$

Answer 1

In order to prove that the geometric definition of the (2-dimensional) dot product is distributive, we use the following diagram:

$\hspace{4.5 cm}$

Note that (whenever $A$ is non-zero) $$ \|B_A\| = \frac{B \cdot A}{\|A\|}\\ \|C_A\| = \frac{C \cdot A}{\|A\|}\\ \|B_A + C_A\| = \frac{(B + C) \cdot A}{\|A\|} $$ It is clear from the diagram that $$ \frac{(B + C) \cdot A}{\|A\|} = \|B_A + C_A\| = \|B_A\| + \|C_A\| = \frac{B \cdot A}{\|A\|}+ \frac{C \cdot A}{\|A\|} $$ the distributivity of the dot-product follows.${}$