A postal address is a list as well: number, street, city. Reorder those three entries and the letter no longer arrives anywhere.

Position one is read across and position two up, so the same two numbers in the other order name a different place. That is the whole force of the word ordered, drawn here in $\mathbb{R}^2$ and true in every $\mathbb{R}^n$.

Fix a positive whole number n. A vector with n components is an ordered list of n real numbers, and the set of all such lists is written $\mathbb{R}^n$. The number sitting in position i is called the i-th component. Two vectors are equal exactly when they have the same number of components and agree in every position.

• $\mathbf{u}=\mathbf{v}$ in $\mathbb{R}^n$ means $u_i=v_i$ for every position i from 1 to n.
• $(1,2)$ and $(2,1)$ are different vectors: position 1 holds 1 in the first and 2 in the second.
• A vector of $\mathbb{R}^2$ is never equal to a vector of $\mathbb{R}^3$, because lists of different lengths cannot agree in every position.
• $\mathbf{0}=(0,\ldots,0)$ is the zero vector of $\mathbb{R}^n$. Each $\mathbb{R}^n$ has its own.

$(2,5)=(2,5)$, and $(2,5)\neq(5,2)$

The second pair holds the same two numbers in exchanged positions, so those vectors are different.

$\{1,1,2\}=\{1,2\}$, and $(1,1,2)\neq(1,2)$

The set forgets the repeated 1. The vector keeps it, and keeps it in position 2.

Two items at a price of thirty. The vector $(30,2)$ would describe thirty items at a price of two.

$\mathbf{0}=(0,0)$ in $\mathbb{R}^2$, and $\mathbf{0}=(0,0,0)$ in $\mathbb{R}^3$

The symbol is written the same way, but the two are different vectors living in different sets.