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.

Position vectors. If $O$ is the origin and $P\in\mathbb R^3$ is any point, the vector $\vec{OP}$ is the position vector of $P$. Every vector $\vec v$ equals the position vector of exactly one point $P$. We reserve the special notations $\hat i,\hat j,\hat k$ for the position vectors of $(1,0,0),(0,1,0),(0,0,1)$ respectively, and $\vec 0$ for the position vector of the origin $(0,0,0)$ — the unique vector of length $0$ (its direction is unspecified / context-dependent). For a point $(a_1,a_2,a_3)$, its position vector is $a_1\hat i+a_2\hat j+a_3\hat k$.

A vector of length $1$ is a unit vector, written $\hat u$; note $\hat i,\hat j,\hat k$ are themselves unit vectors. Real numbers used to scale vectors are called scalars.

Addition and scalar multiplication. For $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$ and a scalar $\alpha\in\mathbb R$:

$\vec a+\vec b=(a_1+b_1)\hat i+(a_2+b_2)\hat j+(a_3+b_3)\hat k,\qquad \alpha\vec a=(\alpha a_1)\hat i+(\alpha a_2)\hat j+(\alpha a_3)\hat k.$ …