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Vector Calculus: An Introduction

Daniel Daners

Section 1.2 Addition and Multiplication by Scalars

Given two vectors \(\vect x=(x_1,\dots,x_N)\) and \(\vect y=(y_1,\dots,y_N)\) in \(\mathbb R^N\) we define their sum by
\begin{equation*} \vect x+\vect y:=(x_1+y_1,\dots,x_N+y_N)\text{.} \end{equation*}
The operation is called the addition of vectors. Moreover, we set
\begin{equation*} -\vect x:=(-x_1,\dots,-x_N) \end{equation*}
and define \(\vect y-\vect x:=\vect y+(-\vect x)\text{,}\) and call it subtraction of vectors. The vector \(-\vect x\) is sometimes called the vector opposite to \(\vect x\). If \(\vect x\) and \(\vect y\) are vectors in the plane or in space the sum and difference can be interpreted geometrically as shown in Figure 1.3.
Figure 1.3. Sum and difference of two vectors.
We define the zero vector in \(\mathbb R^N\) by
\begin{equation*} \vect 0:=(\underset{N \text{ zeros}}{\underbrace{0,\dots,0}})\text{.} \end{equation*}
It has the property that \(\vect x+\vect 0=\vect x\) for all \(\vect x\in\mathbb R^N\text{.}\)
Elements of \(\mathbb R\) are often called scalars (as opposed to vectors). If \(\alpha\) is a scalar and \(\vect x=(x_1,\dots,x_N)\) a vector we define the multiplication by scalars by
\begin{equation*} \alpha\vect x:=(\alpha x_1,\dots,\alpha x_N)\text{.} \end{equation*}
Geometrically this means that we “stretch” the vector \(\vect x\) by the factor \(\alpha\text{.}\)

Note 1.4. Vector space properties.

Note that the algebraic rules for addition and multiplication of \(N\)-vectors are exactly the same as those for vectors in the plane or in space. Given vectors \(\vect x\text{,}\) \(\vect y\) and \(\vect z\) in \(\mathbb R^N\) and scalars \(\alpha,\beta\in\mathbb R\) these rules are:
\begin{align*} \vect x+\vect y\amp=\vect y+\vect x \amp\amp\text{(commutative law)}\\ (\vect x+\vect y)+\vect z\amp=\vect y+(\vect x+\vect z) \amp\amp\text{(associative law)}\\ \vect x+\vect 0\amp=\vect x \amp\amp\text{(existence of a neutral element)}\\ \vect x+(-\vect x)\amp=\vect 0 \amp\amp\text{(existence of an opposite vector)}\\ \alpha(\vect x+\vect y)\amp=\alpha\vect x+\alpha\vect y \amp\amp\text{(distributive law)}\\ (\alpha+\beta)\vect x\amp=\alpha\vect x+\beta\vect x \amp\amp\text{(distributive law)}\\ (\alpha\beta)\vect x\amp=\alpha(\beta\vect x) \amp\amp\text{(associative law)}\\ 1\vect x\amp=\vect x \amp\amp\text{(existence of a neutral element)} \end{align*}
As you learn in linear algebra, these are the axioms of what is usually called a vector space in mathematics.
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