Skip to main content Contents Index Calc Prev Up Next \(\require{cancel}\newcommand{\colorlinecolor}{blue!95!black!30}
\newcommand{\bwlinecolor}{black!30}
\newcommand{\thelinecolor}{\colorlinecolor}
\newcommand{\colornamesuffix}{}
\newcommand{\linestyle}{[thick, \thelinecolor]}
\newcommand{\bbm}{\begin{bmatrix}}
\newcommand{\ebm}{\end{bmatrix}}
\newcommand{\ds}{\displaystyle}
\newcommand{\thet}{\theta}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\vnorm}[1]{\left\lVert\vec #1\right\rVert}
\newcommand{\dotp}[2]{\vec #1 \,\boldsymbol{\cdot}\, \vec #2}
\newcommand{\proj}[2]{\operatorname{proj}_{\,\vec #2}{\,\vec #1}}
\newcommand{\crossp}[2]{\vec #1 \times \vec #2}
\newcommand{\veci}{\vec i}
\newcommand{\vecj}{\vec j}
\newcommand{\veck}{\vec k}
\newcommand{\vecu}{\vec u}
\newcommand{\vecv}{\vec v}
\newcommand{\vecw}{\vec w}
\newcommand{\vecx}{\vec x}
\newcommand{\vecy}{\vec y}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\noin}{\noindent}
\newcommand{\vx}[1][]{\vec{x}_{#1}}
\newcommand{\vxp}{\vec{x}_p}
\newcommand{\vu}{\vec{u}}
\newcommand{\vvv}{\vec{v}}
\newcommand{\vy}{\vec{y}}
\newcommand{\vz}{\vec{z}}
\newcommand{\vb}{\vec{b}}
\newcommand{\vw}{\vec{w}}
\newcommand{\veone}{\vec{e}_1}
\newcommand{\vetwo}{\vec{e}_2}
\newcommand{\vethree}{\vec{e}_3}
\newcommand{\vei}{\vec{e}_i}
\newcommand{\ven}[1]{\vec{e}_{#1}}
\newcommand{\zero}{\vec{0}}
\newcommand{\arref}{\overrightarrow{\text{rref}}}
\newcommand{\tta}{A}
\newcommand{\ttb}{B}
\newcommand{\ttc}{C}
\newcommand{\ttd}{D}
\newcommand{\ttm}{M}
\newcommand{\ttx}{X}
\newcommand{\tti}{I}
\newcommand{\tty}{Y}
\newcommand{\ttp}{P}
\newcommand{\ttat}{A^T}
\newcommand{\ttbt}{B^T}
\newcommand{\ttct}{C^T}
\newcommand{\ttdt}{D^T}
\newcommand{\ttmt}{M^T}
\newcommand{\ttxt}{X^T}
\newcommand{\ttit}{I^T}
\newcommand{\ttyt}{Y^T}
\newcommand{\ttai}{A^{-1}}
\newcommand{\ttbi}{B^{-1}}
\newcommand{\ttxi}{X^{-1}}
\newcommand{\ttpi}{P^{-1}}
\newcommand{\ttaxb}{\tta\vx=\vb}
\newcommand{\ttaxo}{\tta\vx=\zero}
\newcommand{\eyetwo}{\begin{bmatrix}1\amp 0\\0\amp 1\end{bmatrix}}
\newcommand{\eyethree}{\begin{bmatrix}1\amp 0\amp 0\\0\amp 1\amp 0\\0\amp 0\amp 1\end{bmatrix}}
\newcommand{\eyefour}{\begin{bmatrix}1\amp 0\amp 0\amp 0\\0\amp 1\amp 0\amp 0\\0\amp 0\amp 1\amp 0\\0\amp 0\amp 0\amp 1\end{bmatrix}}
\newcommand{\tto}{\textbf{0}}
\newcommand{\lda}{\lambda}
\newcommand{\TT}{[\, T\, ]}
\newcommand{\R}{\mathbb{R}}
\newcommand{\bvm}{\begin{vmatrix}}
\newcommand{\evm}{\end{vmatrix}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Chapter 2 Vectors
This chapter introduces a new mathematical object, the
vector . Defined in
Section 2.2 , we will see that vectors provide a powerful language for describing quantities that have magnitude and direction aspects. A simple example of such a quantity is force: when applying a force, one is generally interested in how much force is applied (i.e., the magnitude of the force) and the direction in which the force was applied. Vectors will play an important role in many of the subsequent chapters in this text.
Until the last section of this chapter we’ll restrict ourselves to vectors in two and three dimensions so that we’re able to understand things visually. However, we’ll also see that the algebraic behaviour of vectors is the same in any dimension, including dimension four or greater. The only thing that changes is the number of coordinates involved. This is one of the great powers of mathematics: we are aided by our visual imagination, but not limited by it.
This chapter begins with moving our mathematics out of the plane and into “space.” That is, we begin to think mathematically not only in two dimensions, but in three. With this foundation, we can explore vectors both in the plane and in space.
