__The Cross Product__

__Definition__

The cross product of __a__ and __b__, written __a__ x __b__, is defined by:

__a__ x __b__ = __n__ a b sin q

where a and b are the magnitude of vectors __a__ and __b__; q is the angle between the vectors, and __n__ is the unit vector (vector with magnitude = 1) that is perpendicular (at 90 degrees to/ orthogonal to/ normal to) both __a__ and __b__. But there are two vectors that this could be - one on either side of the plane formed by the two vectors), so we choose __n__ to be the one which makes (__a__, __b__, __n__) a right handed triad.

Like in the definition of the dot product where we pulled q out of a hat and said it was the angle between the two vectors without any way of finding it, so we need a way of finding __n__ for out definition of the cross product to be any use. Again the __i__, __j__, __k__ vectors come to our rescue, giving us an equivalent definition: let,

__a__ = a_{1} __i__ + a_{2} __j__ + a_{3} __k__

__b__ = b_{1} __i__ + b_{2} __j__ + b_{3} __k__

then,

__a__ x __b__ = ( a_{1} __i__ + a_{2} __j__ + a_{3} __k__) x (b_{1} __i__ + b_{2} __j__ + b_{3} __k__)

The cross product of any two parallel vectors is the null vector since sin 0 = 0, and also

__i__ x __j__ = __k__

__j__ x __k__ = __i__

__k__ x __i__ = __j__

and

__j__ x __i__ = -__k__

__k__ x __j__ = -__i__

__i__ x __k__ = -__j__

Using these, we can eventually find:

__a__ x __b__ = (a_{2}b_{3} - a_{3}b_{2})__i__ + (a_{3}b_{1} - a_{1}b_{3})__j__ + (a_{1}b_{2} - a_{2}b_{1})__k__

That's our equivalent definition. If you're familiar with determinants you may see this can be written more conveniently as,

| i j k|
| a1 a2 a3|
| b1 b2 b3|

__Finding the normal vectors__

Using the equivalence of our two definitions,

__a__ x __b__ = (a_{2}b_{3} - a_{3}b_{2})__i__ + (a_{3}b_{1} - a_{1}b_{3})__j__ + (a_{1}b_{2} - a_{2}b_{1})__k__ = __n__ a b sin q

you can now find __n__

To get used to the cross product try out this cross product applet

The cross product is anticommutative:

__a__ x __b__ = - __b__ x __a__

The cross product of parallel vectors is the null vector, in particular:

__a__ x __a__ = __0__

Also | __a__ x __b__ | is the area of the parallelogram formed by __a__ and __b__,

Area = a b sin q

__Right handed triads__