Talk:PlanetPhysics/Cross Product

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: cross product %%% Primary Category Code: 02. %%% Filename: CrossProcuct.tex %%% Version: 2 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

The cross product or vector product is defined by

$$ \mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$

Like the \htmladdnormallink{dot product}{http://planetphysics.us/encyclopedia/DotProduct.html}, it is useful to look at its geometric definition and properties. Instead of the cosine of the angle between the two \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} the cross product is defined geometrically as

$$ \mathbf{A} \times \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \sin \theta \mathbf{\hat{n}} $$

It is important to see that the \htmladdnormallink{unit vector}{http://planetphysics.us/encyclopedia/PureState.html} $\mathbf{\hat{n}}$ is normal to the plane defined by the two vectors with the direction determined by the right hand rule.

It can be easier to remember the definition of the cross product with the \htmladdnormallink{determinant}{http://planetphysics.us/encyclopedia/Determinant.html} formulation

$$ \mathbf{A} \times \mathbf{B} = \left| \begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$

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