Geometric Introduction to Vectors

A vector $\vec v$ is represented as a directed straight-line segment in 3-dimensional space $\mathbb R^3$: it has an initial point $A=(a_1,a_2,a_3)$ and an end point $B=(b_1,b_2,b_3)$, and is written $\vec{AB}$. The length (magnitude) of $\vec{AB}$ is $|\vec{AB}|=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+(b_3-a_3)^2}$, and its direction is the direction from $A$ to $B$. A vector may be denoted interchangeably by $\vec v$ or $\vec{AB}$.

Equality of vectors. Two vectors $\vec{AB}$ and $\vec{CD}$ in $\mathbb R^3$ are equal, written $\vec{AB}=\vec{CD}$, precisely when the length $|AB|$ equals the length $|CD|$ AND the direction from $A$ to $B$ is parallel to (and the same sense as) the direction from $C$ to $D$. When this holds, $\vec{CD}$ is called a translate of $\vec{AB}$ — every vector can be translated anywhere in space to an equal copy with any chosen initial point. …