Grassmann Algebra

Introduction.nb 5
A plane can be represented by the exterior product of any three points on it, any two points with
a vector parallel to it, or any point on it with a bivector parallel to it.
Graphic showing three views of a plane:
1) Through three points
2) Through two points parallel to a vector
3) Through one point parallel to two vectors
Finally, it should be noted that the Grassmann algebra subsumes all of real algebra, the exterior
product reducing in this case to the usual product operation amongst real numbers.
Here then is a geometric calculus par excellence. We hope you enjoy exploring it.
1.2 The Exterior Product
The anti-symmetry of the exterior product
The exterior product of two vectors x and y of a linear space yields the bivector x�y. The
bivector is not a vector, and so does not belong to the original linear space. In fact the bivectors
form their own linear space.
The fundamental defining characteristic of the exterior product is its anti-symmetry. That is, the
product changes sign if the order of the factors is reversed.
x � y ��y � x
From this we can easily show the equivalent relation, that the exterior product of a vector with
itself is zero.
x � x � 0
This is as expected because x is linearly dependent on itself.
The exterior product is associative, distributive, and behaves as expected with scalars.
Exterior products of vectors in a three-dimensional space
By way of example, suppose we are working in a three-dimensional space, with basis e1 , e2 ,
and e3 . Then we can express vectors x and y as a linear combination of these basis vectors:
2001 4 5
x � a1�e1 � a2�e2 � a3�e3
y � b1�e1 � b2�e2 � b3�e3
1.1
1.2