# Cross product

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You may be looking for Cartesian product.

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. Unlike the dot product, it is only defined in $\R^3$ (that is, three dimensions). It is commonly used in physics, engineering, vector calculus, and linear algebra. It is defined by the formula

$\mathbf a\times\mathbf b=\|\mathbf a\|\|\mathbf b\|\sin(\theta)\,\mathbf n$

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

\begin{align} \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_yb_z-a_zb_y)\mathbf i+(a_zb_x-a_xb_z)\mathbf j+(a_xb_y-a_yb_x)\mathbf k \end{align}

## Properties

• $\vec v\times\vec0=\vec0$
• $\vec v\times\vec v=\vec0$
• $\vec v\times\vec w=-\vec w\times\vec v$
• $a\vec v\times\vec w=a(\vec v\times\vec w)=\vec v\times a\vec w$
• $\vec u\times(\vec v+\vec w)=(\vec u\times\vec v)+(\vec u\times\vec w)$