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Physics: Problems and Solutions

Cross product

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The cross product is one way of taking the product of two vectors (the other being the dot product). This method yields a third vector perpendicular to both. It is defined by the formula

\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta \ \mathbf{n}

where \mathbf{n} is the unit vector perpendicular to both \mathbf{a} and \mathbf{b}. It can be computed other ways as well:

\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
a_x & a_y & a_z\\
b_x & b_y & b_z\\
\end{vmatrix} =

\begin{vmatrix}
a_y & a_z\\
b_y & b_z
\end{vmatrix}\mathbf{i}

-\begin{vmatrix}
a_x & a_z\\
b_x & b_z
\end{vmatrix}\mathbf{j}

+\begin{vmatrix}
a_x & a_y\\
a_x & a_y
\end{vmatrix}\mathbf{k}
= (a_y b_z - a_z b_y)\mathbf{i}+(a_z b_x - a_x b_z)\mathbf{j}+(a_x b_y - a_y b_x)\mathbf{k}

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