Definition

If 2 vectors are in the same $n$-dimensional space, then they can be added. Addition happens cooridnate-wise. In $n$-dimensional space, the formula is the following: $$< a_1, a_2, \dots, a_n > + < b_1, b_2, \dots, b_n > = < a_1 + b_1, a_2 + b_2, \dots, a_n + b_n >$$

For an example in 2-dimensions: $$\begin{align} <2,3>+<4,6> & = <2+4, 3+6> \\ & = <6, 9> \end{align}$$ There is no definition of a sum of 2 vectors of different dimensions.
$$<3,2> + <1,3,4> = \text{undefined}$$

Properties of the Vector Addition

• Commutativity: If $\overrightarrow{x}$ and $\overrightarrow{y}$ are $n$-dimensional vectors, then $$\overrightarrow{x}+\overrightarrow{y}=\overrightarrow{y}+\overrightarrow{x}.$$
• Identity: There is a zero: The vector $(0,0,0,\dots,0)$ with $n$ 0's is the $n$-dimensional $0$ vector. It is written $\overrightarrow{0}.$ For every $n$-dimensional vector $\overrightarrow{x},$ $$\overrightarrow{x}+\overrightarrow{0}=\overrightarrow{x}.$$
• Associativity: For any three $n$-dimensional vectors $\overrightarrow{x},$ $\overrightarrow{y},$ and $\overrightarrow{z},$ $$(\overrightarrow{x}+\overrightarrow{y})+\overrightarrow{z} = \overrightarrow{x}+(\overrightarrow{y}+\overrightarrow{z})$$