Outer Products
Geometric Product
Spacetime vectors are denoted by italic letters to distinguish them from the 3 D vectors
By successive multiplications and additions, the vectors of M 4 generate a geometric algebra G4 = G(M4) called spacetime algebra (STA).
As usual in a geometric algebra, the elements of G4 are called multivectors.
The geometric product uv can be decomposed into a symmetric inner product
u ・ v = 1/ 2 (uv + vu) = v ・ u (5)
and an antisymmetric outer product
u ∧ v = 1/2 (uv − vu) = −v ∧ u (6)
so that
uv = u ・ v + u ∧ v (7)
Outer Product
The outer product along with the notion of k-vector are defined iteratively as follows:
Scalars are defined to be 0-vectors, vectors are 1-vectors, and bivectors, such as u ∧ v, are 2-vectors.
For a given k-vector K, the integer k is called the step (or grade) of K.
For k ≥ 1, the outer product of a vector v with a k-vector K is a (k + 1)-vector defined in terms of the geometric product by
v ∧ K = 1⁄2 (vK + (−1)k Kv) = (−1)k K ∧ v (8)
(k=1) v ∧ w = 1⁄2 (vw − wv) = − w ∧ v
(k=2) v ∧ B = 1⁄2 (v B + B v) = B ∧ v
(k=3) v ∧ T = 1⁄2 (v T - T v) = − T ∧ v
(k=4) v ∧ Q = 1⁄2 (v Q + Q v) = Q ∧ v
Figure 2.1.1: Outer product of a vector v with another vector w
Dynamic app 2.1.1: Outer product of two vectors
Outer product of two vectors (caption for dynamic app 2.1.1)
Drag the green sliders V, W at the right part (projection of vector directions). The red circle is the projection of the bivector V^W.
The right figure shows the corresponding vector (green lines) and bivector (red plane) directions.
Rotate the right figure to check that the bivector plane passes through both vector lines.
Dynamic app 2.1.2: Outer product of a vector and a bivector
Outer product of a vector and a bivector (caption for dynamic app 2.1.2)
The figure shows the projection 3d space of the following (1,3) spacetime elements:
Trivector T (green color sphere). To modify it, drag the black sliders A and Tc.
Bivector B (red color circle). Drag the red point B on the trivector sphere to modify it
Vector V (blue point on the trivector sphere).
The trivector T is the outer product of V and B: T = V^B