35.1: Definition and Forms
- Page ID
- 92264
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The cross product (sometimes called the vector product) is indicated with a cross sign \((\mathbf{A} \times \mathbf{B})\) and is pronounced "A cross B." When you take the cross product of two vectors, you get back another vector, whose magnitude is
\[|\mathbf{A} \times \mathbf{B}|=A B \sin \theta\]
where \(\theta\) is the angle separating vectors \(\mathbf{A}\) and \(\mathbf{B}{ }^{1}\)
The direction of the vector \(\mathbf{A} \times \mathbf{B}\) is perpendicular to the plane of vectors \(\mathbf{A}\) and \(\mathbf{B}\). But there are two possible choices for direction of a vector perpendicular to a plane; which one do we choose? By convention, we choose the one given by a right-hand rule: if you curl the fingers of your right and from vector \(\mathbf{A}\) toward vector \(\mathbf{B}\), then the thumb of your right hand points in the direction of \(\mathbf{A} \times \mathbf{B}\) (Fig. \(\PageIndex{1}\)).
Since \(\mathbf{A} \times \mathbf{B}\) is perpendicular to the plane formed by vectors \(\mathbf{A}\) and \(\mathbf{B}\), it is also perpendicular to both vectors \(\mathbf{A}\) and \(\mathbf{B}\) :\[ (\mathbf{A} \times \mathbf{B}) \perp \mathbf{A} \]
\[ (\mathbf{A} \times \mathbf{B}) \perp \mathbf{B}\]
Component Form
A convenient mnemonic for finding the rectangular components of the cross product is through a matrix determinant:
\[\mathbf{A} \times \mathbf{B} =\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
A_{x} & A_{y} & A_{z} \\
B_{x} & B_{y} & B_{z}
\end{array}\right|\]
\[ =\left(A_{y} B_{z}-A_{z} B_{y}\right) \mathbf{i}-\left(A_{x} B_{z}-A_{z} B_{x}\right) \mathbf{j}+\left(A_{x} B_{y}-A_{y} B_{x}\right) \mathbf{k} .\]
For example, if \(\mathbf{A}=3 \mathbf{i}+5 \mathbf{j}+2 \mathbf{k}\) and \(\mathbf{B}=2 \mathbf{i}-\mathbf{j}+4 \mathbf{k}\), then \(\mathbf{A} \times \mathbf{B}=(20-(-2)) \mathbf{i}-(12-4) \mathbf{j}+(-3-10) \mathbf{k}=\) \(22 \mathbf{i}-8 \mathbf{j}-13 \mathbf{k}\).
Matrix Formulation
Another way to represent the components of the cross product is to write the components of vector \(\mathbf{A}\) into an antisymmetric \(3 \times 3\) matrix, then multiply that matrix by the column vector \(\mathbf{B}\) :
\[\mathbf{A} \times \mathbf{B} =\left(\begin{array}{ccc}
0 & -A_{z} & A_{y} \\
A_{z} & 0 & -A_{x} \\
-A_{y} & A_{x} & 0
\end{array}\right)\left(\begin{array}{c}
B_{x} \\
B_{y} \\
B_{z}
\end{array}\right) \]
\[ =\left(\begin{array}{c}
A_{y} B_{z}-A_{z} B_{y} \\
A_{z} B_{x}-A_{x} B_{z} \\
A_{x} B_{y}-A_{y} B_{x}
\end{array}\right) .
\]
\({ }^{1}\) An old physics joke: What do you get when you cross an elephant with a banana? Ans. "Elephant banana sine \(\theta . "\)