I am trying to write proofs for the following Properties of Vector Addition/Scalar Multiplication in $\mathbb{R}^n$:
Distributive Property 1: $c(\vec{u} + \vec{v}) =c\vec{u} + c\vec{v}$
Distributive Property 2: $(c+d)\vec{u} = c\vec{u} + d\vec{u}$
I have this so far for Distributive Property 1:
$\begin{align} c((u_{1}, u_{2} , ... , u_{n}) +(v_{1}, v_{2}, ... , v_{n})) && \text{Definition of Vectors in n-dimensions}\tag1\\ c(u_{1} + v_{1}, u_{2} + v_{2}, ... , u_{n} + v_{n}) && \text{Vector Addition}\tag2\\ (c(u_{1} + v_{1}), c(u_{2} + v_{2}), ... , c(u_{n} + v_{n})) && \text{Scalar Multiplication}\tag3\\ (cu_{1} + cv_{1}, cu_{2} + cv_{2}, ... , cu_{n} + cv_{n}) && \text{Distributive Property of } \mathbb{R}\tag4\\ (cu_{1}, cu_{2}, ... ,cu_{n}) +(cv_{1},cv_{2},...,cv_{n}) && \text{Vector Addition (Reverse)}\tag5\\ c(u_{1}, u_{2}, ... , cu_{n}) + c(v_{1}, v_{2},...,v_{n}) && \text{Scalar Multiplication (Reverse)}\tag6\\ \end{align}$
However, I am having trouble discerning the difference between Distributive Property of Real Numbers and Scalar Multiplication and knowing which one to use/cite in my proofs.
On line 3, I originally had Distributive Property of Real Numbers as opposed to Scalar Multiplication, but my professor corrected it to Scalar Multiplication.
Similarly, I have the same concerns for Distributive Property 2:
$\begin{align} (c+d)(u_{1}, u_{2}, ... , u_{n}) && \text{Definition of Vectors in n-dimensions} \tag1\\ ((c+d)u_{1}, (c+d)u_{2}, ... , (c+d)u_{n}) && \text{Scalar Multiplication}\tag2\\ ((cu_{1} + du_{1}, cu_{2} + du_{2}, ... , cu_{n} + du_{n}) && \text{Scalar Multiplication}\tag3\\ (cu_{1}, cu_{2}, ... , cu_{n}) + (du_{1}, du_{2}, ... ,du_{n}) && \text{Vector Addition (Reverse)}\tag4\\ \end{align}$
I recall my professor saying about how it is due to the differences between whether or not the scalar is applied on real numbers and vectors; in this example, I can see how that would relate to line 2, as we are applying a scalar on a vector. However, given that $(c+d)$ is now applied to all the elements of the vector, would we not begin to use the distributive property on line 3?