Let

**A**= (

*a*

_{1},

*a*

_{2},

*a*

_{3}),

**B**= (

*b*

_{1},

*b*

_{2},

*b*

_{3}). Let A and B be the magnitudes of

**A**and

**B**, respectively, and let

*[itex]\theta[/itex]*be the angle between the vectors.

**A**X

**B**= AB sin(

*[itex]\theta[/itex]*)

**A**X

**B**= (

*a*

_{2}

*b*

_{3}-

*a*

_{3}

*b*

_{2},

*a*

_{3}

*b*

_{1}-

*a*

_{1}

*b*

_{3},

*a*

_{1}

*b*

_{2}-

*a*

_{2}

*b*

_{1})

As I understand it, the cross product was developed for application to situations in physics, in which two vectors produce a result perpendicular to both of them. Therefore, the cross product would be orthogonal to both vectors: A [itex]\cdot[/itex] (

**A**X

**B**) = 0, and B [itex]\cdot[/itex] (

**A**X

**B**) = 0. However, I do not understand the other properties of the cross product which led to these definitions. Thank you for your help.