# Anyone have a clear definition of how to do a cross product?

1. Nov 7, 2004

### Rocko

anyone have a clear definition of how to do a cross product?

2. Nov 7, 2004

### Galileo

The cross-product $\vec C$ of two vectors $\vec A$ and $\vec B$ is most conveniently defined by:
$|\vec C|=|\vec A||\vec B|\sin(\theta)$
where $\theta$ is the angle between $\vec A$ and $\vec B$.
This gives the magnitude of $\vec C$. The direction is given by the right-hand rule.

To calculate the cross product when you know the components,
it's usually easiest to form the symbolic 3X3 determinant:
$$\vec A=(A_x,A_y,A_z)$$
$$\vec B=(B_x,B_y,B_z)$$
$$\vec C = \left| \begin{array}{ccc} \hat x & \hat y &\hat z \\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{array}\right|$$
This follows from $\hat x \times \hat y = \hat z$ (and the other possible product combinations of these unit vectors) and the distributivity of the cross product (which is tedious to prove IIRC).