# Maths - 2D Euclidean Space - Outer Product

For the 1D case on this page we calculated the full table for the outer product for null basis, but here the full table is a bit big so I have put it at the bottom of this page, here we just have the entries for cross multiplying two vectors:

 a^b b.n0 b.n∞ b.n1 b.n2 a.n0 0 n0∞ -1/2 n01 n02 a.n∞ -n0∞+ 1/2 0 n∞1 n∞2 a.n1 -n01 -n∞1 0 n12 a.n2 -n02 -n∞2 -n12 0

Each term in this table is calculated from:
n0= (e1 + e2)/2
n= (e1 - e2)/2
n1= e3
n2= e4

So we just multiply out each term by converting to 'e' basis, doing inner product, then converting back to 'n' basis.

## The outer product of two null vectors

Lets take the point 'p' (x1,y1) in Euclidean space, this gives,

p=(-1,x1²+y1²,x1,y1)

and we want to take the outer product with q:

q=(-1,x2²+y2²,x2,y2)

So, multiplying out the terms using the above table, the outer product is the multivector:

scalar = (x1²+y1² -x2²- y2²)/2
n0∞ = x2²+y2² - x1²-y1²
n01 = x1 - x2
n02 = y1 - y2
n∞1 = x1²x2 - x2²x1 = x1x2(x1 - x2)
n∞y = y1²y2 - y2²y1 = y1y2(y1 - y2)
n12 = x1 y2 - x2 y1

## Meet

In the above example if x1=x2 and y1=y2 then:

scalar = 0
n0∞ = 0
n01 = 0
n02 = 0
n∞1 = 0
n∞y = 0
n12 = 0

So this gives us a way to test if x1 and x2 represent the same point since, if they do then,

p^q=0

 a*b b.n b.n0 b.n∞ b.n0∞ b.n1 b.n01 b.n∞1 b.n0∞1 b.n2 b.n02 b.n∞2 b.n0∞2 b.n12 b.n012 b.n∞12 b.n0∞12 a.n 1 n0 n∞ n0∞ n1 n01 n∞1 n0∞1 n2 n02 n∞2 n0∞2 a.n0 n0 0 n0∞ -1/2 n0/2 n01 0 n0∞1-n1/2 n01/2 n02 0 n0∞2-n2/2 0 a.n∞ n∞ -n0∞+ 1/2 0 n∞/2 n∞1 -n0∞1+n1/2 0 n∞1/2 n∞2 n2/2-n0∞2 0 n∞2/2 a.n0∞ n0∞ n0/2 n∞/2 n0∞-1/4 n0∞1 n01/2 n∞1/2 n0∞1-n1/2 n0∞2 n02/2 n∞2/2 n0∞2-n2/2 a.n1 n1 -n01 -n∞1 n0∞1 0 0 0 0 0 0 0 0 a.n01 n01 0 -n0∞1+n1/2 n01/2 0 0 0 0 0 0 0 0 a.n∞1 n∞1 n0∞1-n1/2 0 n∞1/2 0 0 0 0 0 0 0 0 a.n0∞1 n0∞1 n01/2 n∞1/2 n0∞1-n1/2 0 0 0 0 0 0 0 0 a.n2 n2 -n02 -n∞2 n0∞2 0 0 0 0 0 0 0 0 a.n02 n02 0 -n0∞2+n2/2 n02/2 0 0 0 0 0 0 0 0 a.n∞2 n∞2 n0∞2-n2/2 0 n∞2/2 0 0 0 0 0 0 0 0 a.n0∞2 n0∞2 n02/2 n∞2/2 n0∞2-n2/2 0 0 0 0 0 0 0 0 a.n12 0 0 0 0 0 0 0 0 0 0 0 0 a.n012 0 0 0 0 0 0 0 0 0 0 0 0 a.n∞12 0 0 0 0 0 0 0 0 0 0 0 0 a.n0∞12 0 0 0 0 0 0 0 0 0 0 0 0