Vector Calculus: An Introduction

Given two vectors $\mathit{x}=(x_1,\dots ,x_N)$ and $\mathit{y}=(y_1,\dots ,y_N)$ in ${\mathbb{R}}^N$ we define their sum by

The operation is called the addition of vectors. Moreover, we set

and define $\mathit{y}-\mathit{x}:=\mathit{y}+(-\mathit{x})\text{,}$ and call it subtraction of vectors. The vector $-\mathit{x}$ is sometimes called the vector opposite to $\mathit{x}$. If $\mathit{x}$ and $\mathit{y}$ are vectors in the plane or in space the sum and difference can be interpreted geometrically as shown in Figure 1.3.

We define the zero vector in ${\mathbb{R}}^N$ by

It has the property that $\mathit{x}+\mathbf{0}=\mathit{x}$ for all $\mathit{x}\in {\mathbb{R}}^N\text{.}$

Elements of $\mathbb{R}$ are often called scalars (as opposed to vectors). If $\alpha$ is a scalar and $\mathit{x}=(x_1,\dots ,x_N)$ a vector we define the multiplication by scalars by

Geometrically this means that we “stretch” the vector $\mathit{x}$ by the factor $\alpha \text{.}$