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Mathematics LibreTexts

4.1: Addition and Scalar Multiplication in \(\mathbb{R}^{n}\)

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    A simple but important property of \(n\)-vectors is that we can \(\textit {add}\) \(n\)-vectors and \(\textit {multiply}\) \(n\)-vectors by a scalar:


    Given two \(n\)-vectors \(a\) and \(b\) whose components are given by 

    $$a=\begin{pmatrix}a^{1} \\ \vdots  \\ a^{n}\end{pmatrix} ~and ~b=\begin{pmatrix}b^{1} \\ \vdots  \\ b^{n}\end{pmatrix}$$ their \(\textit{sum}\) is

    $$a+b := \begin{pmatrix} a^1+b^1 \\ \vdots \\ a^n+b^n\end{pmatrix}\, .$$  

    Given a scalar \(\lambda\), the \(\textit{scalar multiple}\)

    $$\lambda a := \begin{pmatrix}\lambda a^{1} \\ \vdots \\ \lambda a^{n}\end{pmatrix}\, .$$

    Example 43


    $$a=\begin{pmatrix}1\\2\\3\\4\end{pmatrix} ~and ~b = \begin{pmatrix}4\\3\\2\\1\end{pmatrix}\, .$$

    Then, for example,

    $$a+b= \begin{pmatrix}5\\5\\5\\5\end{pmatrix} and 3a - 2b= \begin{pmatrix}-5\\0\\5\\10\end{pmatrix}\, .$$

    A special vector is the \(\textit{zero vector}\). All of its components are zero:

    $$0=\begin{pmatrix}0\\ \vdots \\ 0\end{pmatrix}\, .$$

    In Euclidean geometry---the study of \(\mathbb{R}^{n}\) with lengths and angles defined as in section 4.3---\(n\)-vectors are used to label points \(P\) and the zero vector labels the origin \(O\). In this sense, the zero vector is the only one with zero magnitude, and the only one which points in no particular direction.