Cross Product

The Cross Product is a property of vectors. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.

Definition

If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to |u||v|sin(θ). Thus,

|u × v| = |u||v||sin(θ)||n| 
|u × v| = |u||v||sin(θ)|

In other applications, this magnitude has been found to be equal to the area of the parallelogram whose vertecies are at 0, u, u+v, and v.

In addition,

u × 0 = 0
u × v = -(v × u)
(ru) × (sv) = (rs) u × v (where r and s are scalars)

General Application of the Cross Product

It is common practice in mathematics to write vecotrs as sums of th unti vectors, i, j, and k. These each have a magnitude of one and, from the origin, move precisely 1 unit on the x, y, and z axes respectively.

Say, then, that we have two vectors, u and v.

u = <u1,u2,u3>
v = <v1,v2,v3>

We can rewrite these vectors in terms of i, j, and k.

u = u1i + u2j + u3k
v = v1i + v2j + v3k

We can then "multiply" them in an algebraic manner.

u × v = (u1v1)i × i + (u1v2)i × j + (u1v3)i × k + (u2v1)j × i + (u2v2)j × j + 
        (u2v3)j × k + (u3v1)k × i + (u3v2)k × j + (u3v3)k × k

We can remove immediately any terms where the vector is crossing itself. The angle between two parallel vectors, a and b, is 0, thus a × b = |a||b|sin(0)|n| = 0 . So, the above statement simplifies to

u × v = (u1v2)i × j + (u1v3)i × k + (u2v1)j × i + (u2v3)j × k + (u3v1)k × i + (u3v2)k × j

Because the cross product reflects the normal vector, the following hold:

i × j = -(j × i) = k
j × k = -(k × j) = i
k × i = -(i × k) = j

Then, we can simplify our equation further:

u × v = (u1v2)k + (u1v3)(-j) + (u2v1)(-k) + (u2v3)i + (u3v1)j + (u3v2)(-i)
      = (u2v3-u3v2)i + (u3v1-u1v3)j + (u1v2-u2v1)k
      = (u2v3-u3v2)i - (u1v3-u3v1)j + (u1v2-u2v1)k (u-components are  in order)

This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in matricies

=

u2 u3 v2 v3

i -

u1 u3 v1 v3

j +

u1 u2 v1 v2

k

=	i j k u1 u2 u3 v1 v2 v3

The Cross Product in GraalScript

The cross product of two vector arrays in Graal can be obtained through the vectorcross(u,v) function. It's return value is a vector, w₁i + w₂j + w₃k, where the vector w is calculated from the determinant of the above matrix.