Addition, Subtraction, and Scalar Multiplication of Real 2D Vectors (R^2)

Suppose we have two vectors $\vec{v} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$ and $\vec{w} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$.

What is the meaning of $\vec{v} + \vec{w}$ and how do we calculate it?

Visually, we can add vectors tip-to-tail as such:

We can see from the plot that $\vec{v} + \vec{w} =  \begin{bmatrix} 5 \\ 5 \end{bmatrix}$. 

Mathematically, we add vectors component by component, i.e.

$$\begin{align*} \vec{v} + \vec{w}  &= \begin{bmatrix} 1 \\ 3 \end{bmatrix} +  \begin{bmatrix} 4 \\ 2 \end{bmatrix} \\ &= \begin{bmatrix} 1 + 4 \\ 3 + 2 \end{bmatrix}  \\ &= \begin{bmatrix} 5 \\ 5 \end{bmatrix}. \end{align*}$$

In general, for vectors $\vec{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ and $\vec{w} = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ we have:

$$\begin{align*} \vec{v} + \vec{w}  &= \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} +  \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \\ &= \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2  \end{bmatrix}. \end{align*}$$

...