...or is there more to the story of vector multiplication than is commonly shared with students in an introductory physics course?

**Table of Contents:**

- Two dimensions and the right-angle unit-bivector
**i** - Reciprocal space and the
*Cartesian unit-trivector* - Space-time and the
*Minkowski pseudoscalar* - Read more about it.

and the gate is so low and small

that one can only enter it as a little child."

Vectors **a** and **b** are *added* according to the parallelogram rule,
labeled in the figure below (following Roger Penrose) as though it is also
the rule for adding complex numbers. We've added color to the images to show
further how the hue and lightness of a single pixel can be used to uniquely specify a
position on the 2D manifold, and then some.

Emboldened by the above image, we might then
visualize the mulitvector *product* of **a** and **b** as
a directed-arc on a circle of radius ab.
This product can be written as
**ab** = **a**⋅**b** + **a**^**b** =
ab Cos[φ_{ab}] + **i** ab Sin[φ_{ab}] =
ab e^{i φab} where φ_{ab} is the
*angle* between **a** and **b**. Here **a**⋅**b** is the
*dot* or *scalar-product* of
**a** and **b**, and **a**^**b** is the *directed-area
bivector*
for **a** and **b**. This latter is like the *cross-product*
**a**×**b** (same size and
anti-commutivity), except that as a directed-area it "lives in" the
plane defined by **a** and **b**. The quantity **i** is the
right-angle unit-bivector, whose product with itself is the scalar
quantity -1 since "two right angles is a U-turn".
The quantity e^{i φab} is a *rotor*
(a unit-circle directed-arc or unit-multivector)
which rotates whatever it multiplies by φ_{ab}.

As you can see from the Penrose visualization
below for graphically constructing the complex number (or multivector) product
**ab**, the product of vector **a** with a unit vector in the "real"
direction (a special case of the product relation above) is just the
complex number |**a**|e^{iφ}, where φ is the
angle that **a**
makes with that unit vector. In this way bivectors
emerge from vector multiplication in 2D as *imaginary numbers*, which
in that disguise have long been helping folks put 2D vectors to use
with no mention of vector products *per se*.
Phasor diagrams for the study of AC circuits, and phase-amplitude analysis in
the study of spring oscillations and wave optics, are places that you might
encounter such quantities in an intro-physics course. Historians may
thus say that addition and multiplication of numbers in the
"Wessel Argand Gauss complex plane" looks a lot like addition and
multiplication of *vectors in xy*. Perhaps they are one and the same.

Does this mean that 2D vectors
and complex numbers are also one and the same? **No.**
Since scalars, vectors,
and bivectors are of grade 0, 1, and 2 respectively,
*vector products* in a plane span the *even subalgebra* (namely
scalars and bivectors, or reals and imaginaries) of Euclidean two-space.
Vectors in xy span the *odd subalgebra*.
As indicated above, however, multiplication of any
vector by a unit reference vector moves it into the even subalgebra,
which then acts oddly like a collection of vectors in xy.
Below we discuss how such *subalgebra mimicry* makes
tape-measure and time-piece industries quite separate from one another
in the (3+1)D world that we live in as well.

In Euclidean 3-space, one can apply
the concepts above on a suitably-oriented planar surface. In that sense the
*vector product* **a** and **b** can still be seen as a
directed-arc on a planar circle of radius ab. Alternatively,
one can use the right-handed orthogonal unit-vectors of a
*standard Cartesian frame* {**σ**_{x},
**σ**_{y},
**σ**_{z}} to define a Cartesian unit-trivector *i*_{3}
as the *directed-volume* vector-product
**σ**_{x}**σ**_{y}**σ**_{z}.
The bivector (or covector) basis in that case can be written as:
**σ**_{x}**σ**_{y} = *i*_{3} **σ**_{z},
**σ**_{y}**σ**_{z} = *i*_{3} **σ**_{x}, and
**σ**_{z}**σ**_{x} = *i*_{3} **σ**_{y}.
Thus the Cartesian unit-trivector *i*_{3} lets us define
a dual vector-space (a vector set mapped to the bivector or one-form basis)
that proves helpful e.g. in the analysis of: gradients, angular momentum,
particle scattering, stress/strain, quantized spin,
and crystal periodicity. In fact one can show that
*nature overtly expresses her fondness* for
such reciprocal or dual spaces in everything from the uncertainty
principle (e.g. linking transverse momentum spread in diffraction
with our knowledge of "which slit") to the relativistic relationship
between time and space (see the next section).

The Cartesian unit-trivector lets us
also write the vector product above in 3 dimensions
as **ab** = **a**⋅**b** + **a**^**b** = **a**⋅**b**
+ *i*_{3} **a**×**b**. The familiar *cross-product*
**a**×**b** (figure at right) is therefore the
3-vector dual to the bivector (or directed-area) **a**^**b**.
Hence in crystallography, for example, the reciprocal-lattice vector
basis {**a***,**b***,**c***} is simply written
in terms of direct-lattice
basis vectors {**a**,**b**,**c**} via
relations like **a*** = (**b**×**c**)/V_{c}
and **b*** = (**c**×**a**)/V_{c},
where the unit cell volume V_{c} = **a**⋅(**b**×**c**).
Use of cross-products to describe e.g. torques and
magnetic forces in intro-physics courses also relies on
the idea of vectors that are orthogonal, and in this sense dual,
to a directed area. It's helpful to know this, because not
all problems involving a directed area can be solved from
properties of the cross-product alone. For example bivectors
transform differently than vectors, and the wobble of a
lopsided football might involve a matrix rather than a scalar
moment of inertia.

As with the right-angle
unit-bivector **i**, when multiplied by itself the Cartesian
unit-trivector *i*_{3} also equals the scalar quantity -1.
Hence in vector geometry the square root of minus one has
more than one important, and non-imaginary, place to call home.

Vector multiplication in spacetime at
the very least makes contact with intro-physics in so far as
gravity and magnetism are relativistic effects.
To explore this, we start with the *oriented unit-hypervolume* named
in the section title. This dude is the 4D version of "two right turns in a plane".
Hence it too yields "-1" when multiplied by itself.

For example, a right-handed
set of orthogonal unit-vectors in flat-spacetime {γ_{t},
γ_{x},
γ_{y},
γ_{z}} lets one define the Minkowski pseudoscalar
*I*_{4} as the quadvector product
γ_{t}γ_{x}γ_{y}γ_{z}.
The time-vector γ_{t} in this basis-set *defines
simultaneity* for a subset of co-moving objects, which after
*multiplication by any other spacetime vector* (e.g. displacement,
or energy-momentum) describes the space-time
split in that vector (time vs. space, or mass-energy vs. momentum) as
it is experienced by those co-moving
objects. In other words, multiplication with
an object's time unit-vector γ_{t} immediately
tosses spacetime vectors into an even subalgebra (or spacetime split)
whose scalar, bivector and quadvector elements (reminescent of
the 2D case above) masquerade
beautifully as a set of spatial 3-vectors parameterized by object
scalar time. **This is the world that each of us experiences.**

The (2+1)D figure
at right plots an accelerated object's time
in the vertical direction, and its position in the cyan t=0 plane (xct version
here).
The little gray spaceship
in the center (with orange exhaust) accelerates at 1 gee for 2 years according
to onboard clocks, then decelerates for two more so as to pick up
some specimens in the red dwarf solar system of Barnard's star
5.52 lightyears away (at the future time of this voyage) before
heading back home. The roundtrip takes about 14.5 years on earth clocks.
This shows that the relative place and time of events (colored points) in a
space-time split changes significantly with the direction of
one's time unit vector γ_{t}. Earth's view of the
same trip is
here.

As the animation shows, traveling over 5.5
lightyears in 4 traveler years is facilitated by the spaceship's ability
to "length-contract the distance" between
Earth and Barnard's star within its space-time split. Intersection
between the t=0 plane and the dotted magenta world line marks an interesting
location whose distance to the spaceship remains constant during the 1st
and 4th quarters of the trip (only in the traveler's split) because the
effects of acceleration and contraction cancel out.
The energy required for the trip is
another
story. The top and bottom
surfaces
of earth's blue lightcone track planes of constant elapsed time on earth.
In spite of all the rearrangement
going on, the solid black curved world line of the
traveler stays within the traveler's yellow future and past light
cones. Note also that the *proper-time* elapsed along the traveler's
world line, like the *proper-distance* between the Earth and red dwarf
in their common frame of motion, are "minimally-variant" quantities of
more general interest than the space-time split between events and
world-lines from other points of view.

Vector products also have many other uses.
In the case of displacements, for instance, the
time-vector product times its conjugate yields the *metric equation*
or space-time version of Pythagoras' theorem
(*c*dτ)^{2} = (*c*dt)^{2} - (d**x**)^{2}.
This relates the proper time dτ elapsed between two events in the life
of a traveler, to the time (dt) and space (d**x**) intervals
measured on a map frame of interconnected yardsticks and synchronized
clocks. Part-per-billion shifts in the metric equation in turn give
rise to gravity on earth. If we instead multiply γ_{t} by
space-time's electromagnetic field bivector, we find how this bivector
breaks down (for those co-moving objects) into separate electric and magnetic
field components. This breakdown can be written using the Minkowski pseudoscalar
as **E** + *I*_{4} *c***B**, which in turn helps explain
quantitatively how moving charges give rise to magnetic fields as a result
of the finite lightspeed constant *c* that connects time
and space.

- These notes on the geometric home of imaginary
numbers were inspired by
D. Hestenes
(2003)
*Am. J. Phys.***71**(105-121). - The first two figures
were adapted from Figure 2.3 of R. Penrose (1997)
*The large, the small, the human mind*(Cambridge U Press).

This page is http://www.umsl.edu/~fraundorfp/p001htw/vectorProducts.html. Although there are many contributors, the person responsible for errors is P. Fraundorf. This site is hosted by the