# 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 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 = a1 i + a2 j + a3 k
b = b1 i + b2 j + b3 k
then,
a x b = ( a1 i + a2 j + a3 k) x (b1 i + b2 j + b3 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 = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)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|
```

### Some Properties of the Cross Product

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