Euclidean vector

Definition

Let $n$ be a positive integer. A Euclidean vector in the Euclidean space $\mathbb{R}^n$ is an element $v = (v_1, \dots, v_n)$ of $\mathbb{R}^n$. The real numbers $v_1, \dots, v_n$ are the components of $v$. For two points $A = (a_1, \dots, a_n)$ and $B = (b_1, \dots, b_n)$, the vector from $A$ to $B$ is $$\overrightarrow{AB} = (b_1 - a_1, \dots, b_n - a_n).$$ The sum of two vectors $u$ and $v$ is $u + v = (u_1 + v_1, \dots, u_n + v_n)$. The product of a real number $c$ and a vector $v$ is $c v = (c v_1, \dots, c v_n)$. The zero vector is $0 = (0, \dots, 0)$.

Remarks

Two arrows give the same vector exactly when a translation moves one arrow onto the other. For each three points $A$, $B$ and $C$, the vectors satisfy $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$. With the sum and the product by real numbers, the vectors of $\mathbb{R}^n$ form a real vector space of dimension $n$.

Explanations

L1 human

A vector tells you how to move: how far, and in which direction. The vector $(3, 4)$ in the plane says "move 3 to the right and 4 up". It does not matter where you start, so arrows of the same length and direction are the same vector.