Inner Products
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)
The inner product of a vector v with a multivector K is defined by
v ・ K = 1⁄2 (vK + (−1)k+1 Kv) = (−1)k+1 K ・ v (9)
(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
and it can be proved that the result is a (k −1) -vector.
Orthogonal vectors and signatures
The product of two mutually orthogonal vectors is zero.
The set of vectors whose inner product with a given vector is zero belong to its orthogonal multivector direction.
The inner product of a vector direction with itself is called its signature.
The convention (1,n) means that timelike vectors have positive signature whilst spacelike vectors have negative signature.
Between both sets lies the so-called null-cone, whose vectors have a null signature.
Dynamic app 2.2.1: Orthogonal directions
Orthogonal bivector in (1,2) spacetime (caption for dynamic app 2.2.1)
The green point at the left side is the proyection of a certain spacetime vector direction V, which can be changed by dragging.
The red circumference is the set of points whose projected vector spacetime directions are orthogonal to the direction V.
At the right side, the relation between the red plane (which is orthogonal to the green line) can be understood as a reflection through the light cone (dashed line).
This orthogonality is radically different from the euclidean case, where the red plane would be perpendicular to the green line.
Placing the green point V on any of the null circumferences (blue for the Future cone and violet for the Past cone, which are also shown at the right figure with these same colors), it is also coincident with the red circumference.
In this case, the vector direction is orthogonal to itself, which is another way to understand the concept of null signature.
Positive and negative inner products
The orthogonal plane to a given vector V separates spacetime into two regions, one with vectors with a positive inner product with V and the other one with negative inner products.
Figure 2.2.1: Inner product with a spacelike vector
Figure 2.2.2: Inner product with a timelike vector
Disclaimer
Every point on the projection plane represents all colinear vectors to a given one, even those opposite to it.
For this reason, the signs on the projection space showed previously correspond only to the side "looking to V", whereas the vectors with are "looking against V" would have the opposite sign.