13.1: Vectors
- Page ID
- 47300
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition
We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms of its components with respect to a reference system as
\[ \vec{a} \, = \, \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix}. \]
The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
- Vector addition: \begin{align} \vec{a} + \vec{b} \, &= \, \vec{c}\\[4pt][5 pt] \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix} + \begin{Bmatrix} \,3\, \\[4pt] \,3\, \\[4pt] \,2\, \end{Bmatrix} \, &= \, \begin{Bmatrix} \,5\, \\[4pt] \,4\, \\[4pt] \,9\, \end{Bmatrix} \end{align} Graphically, addition is stringing the vectors together head to tail.
- Scalar multiplication: \[ -2 \times \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix} \, = \, \begin{Bmatrix} \,-4\, \\[4pt] \,-2\, \\[4pt] -14 \end{Bmatrix} \]
\(\PageIndex{2}\): Vector Magnitude
The total length of a vector of dimension \(m\), its Euclidean norm, is given by
\[ || \vec{x} || \, = \, \sqrt{\sum_{i=1}^{m} x_i^2}. \] This scalar is commonly used to normalize a vector to length one.
\(\PageIndex{3}\): Vector Dot or Inner Product
The dot product of two vectors is a scalar equal to the sum of the products of the corresponding components: \[\vec{x} \cdot \vec{y} \, = \, \vec{x}^T \vec{y} \, = \, \sum_{i=1}^{m} x_i y_i.\] The dot product also satisfies \[\vec{x} \cdot \vec{y} \, = \, ||\vec{x}|| ||\vec{y}|| \cos \theta, \] where \(\theta\) is the angle between the vectors.
\(\PageIndex{4}\): Vector Cross Product
The cross product of two three-dimensional vectors \(\vec{x}\) and \(\vec{y}\) is another vector \(\vec{z}\), written as \(\vec{x} \times \vec{y} = \vec{z}\). Vector \(\vec{z}\)'s
- direction is normal to the plane formed by the other two vectors,
- direction is given by the right-hand rule, rotating from \(\vec{x}\) to \(\vec{y}\),
- magnitude is the area of the parallelogram formed by the two vectors – the cross product of two parallel vectors is zero – and
- (signed) magnitude is equal to \(||\vec{x}|| ||\vec{y}|| \sin \theta\), where \(\theta\) is the angle between the two vectors, measured from \(\vec{x}\) to \(\vec{y}\).
In terms of their components,
\[ \vec{x} \times \vec{y} \, = \, \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\[4pt] x_1 & x_2 & x_3 \\[4pt] y_1 & y_2 & y_3 \end{vmatrix} \, = \, \begin{Bmatrix} (x_2 y_3 - x_3 y_2) \hat{i} \\[4pt] (x_3 y_1 - x_1 y_3) \hat{j} \\[4pt] (x_1 y_2 - x_2 y_1) \hat{k} \end{Bmatrix}. \]