Intuitive proof for distributive property of dot product using $\overrightarrow{u}\cdot\overrightarrow{v} = u_{x}v_{x} + u_{y}v_{y}$

Question

I understand the intuitive way to think of the distributive property using $\overrightarrow{A}\cdot\overrightarrow{B} = AB\cos\theta$:

Then, this makes me wonder if it's possible to prove distributive property using the more general form for dot product: $\overrightarrow{u}\cdot\overrightarrow{v} = u_{x}v_{x} + u_{y}v_{y}$

Answer 1

It is much simpler using the coordinate definition because it just comes down to the distributivity of real multiplication. If you write $R_x=B_x+C_x$ and the same for $y$, then $A\cdot R=A_xR_x+A_yR_y=A_x(B_x+C_x)+A_y(B_y+C_y)$ and now use distributivity and associativity in the reals.