12.4 Polyvectors

We've now come to the key definition that splits our theory of vectors and rulers away from our theory of metrics!

Definition.

The product of vectors is anti-commutative

\[ \begin{align} w \mult v &= - v \mult w \end{align} \]

and the square of a vector is zero

\[ \begin{align} v \mult v &= 0 \end{align} \]

where \( v : \Vector_{} \) and \( w : \Vector_{} \) are any vectors.

Applied to the standard vectors, we have:

\[ \begin{align} \vec{\standard}_{x} \mult \vec{\standard}_{x} &= 0, & \vec{\standard}_{y} \mult \vec{\standard}_{x} &= - \vec{\standard}_{x} \mult \vec{\standard}_{y}, & \vec{\standard}_{y} \mult \vec{\standard}_{y} &= 0. \end{align} \]

With these multiplication laws in place, we can talk about polyvectors.

Definition.

Numbers are polyvectors with degree zero, the standard vectors,

\[ \begin{align} \vec{\standard}_{x} &: \PolyVector_{(x, y)}^{1} & &\andSpaced & \vec{\standard}_{y} &: \PolyVector_{(x, y)}^{1}, \end{align} \]

are polyvectors with degree one, and all other polyvectors are made from these by adding and multiplying.

Let's get a little practice with polyvectors by multiplying two vectors together.

Example.

Let's multiply the vectors

\[ \begin{align} v &: \Vector_{(x, y)} & & & w &: \Vector_{(x, y)} \\ v &= \vec{\standard}_{x} - 4 \mult \vec{\standard}_{y} & &\andSpaced & w &= 2 \mult \vec{\standard}_{x} + \vec{\standard}_{y}. \end{align} \]

We compute.

\[ \begin{align} v \mult w &= \bigl(\vec{\standard}_{x} - 4 \mult \vec{\standard}_{y}\bigr) \mult \bigl(2 \mult \vec{\standard}_{x} + \vec{\standard}_{y}\bigr) \\ &= 2 \mult \vec{\standard}_{x} \mult \vec{\standard}_{x} + \vec{\standard}_{x} \mult \vec{\standard}_{y} - 8 \mult \vec{\standard}_{y} \mult \vec{\standard}_{x} - 4 \mult \vec{\standard}_{y} \mult \vec{\standard}_{y} \\ &= \vec{\standard}_{x} \mult \vec{\standard}_{y} - 8 \mult \vec{\standard}_{y} \mult \vec{\standard}_{x} \\ &= \vec{\standard}_{x} \mult \vec{\standard}_{y} + 8 \mult \vec{\standard}_{x} \mult \vec{\standard}_{y} \\ &= 9 \mult \vec{\standard}_{x} \mult \vec{\standard}_{y} \end{align} \]

The result is a polyvector with degree two!

Polyvector Standard Forms

Using our multiplication laws, we can rewrite polyvectors into standard forms.

degree	polyvector standard form
0	\( a \)
1	\( a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y} \)
2	\( a \mult \vec{\standard}_{x} \mult \vec{\standard}_{y} \)

Like polymetrics, polyvectors in degrees zero and one are familiar objects.

Important.

A degree zero polyvector is a number and a degree one polyvector is a vector.

Unlike polymetrics, polyvectors for the \( (x, y) \)-coordinate plane exist only in degrees zero, one, and two. In degree three, each of the possible products

\[ \begin{align} &\vec{\standard}_{x} \mult \vec{\standard}_{x} \mult \vec{\standard}_{x}, & &\vec{\standard}_{x} \mult \vec{\standard}_{x} \mult \vec{\standard}_{y}, & &\vec{\standard}_{x} \mult \vec{\standard}_{y} \mult \vec{\standard}_{y}, & &\vec{\standard}_{y} \mult \vec{\standard}_{y} \mult \vec{\standard}_{y} \end{align} \]

either contains \( \vec{\standard}_{x} \mult \vec{\standard}_{x} \) or \( \vec{\standard}_{y} \mult \vec{\standard}_{y} \) and is therefore zero. To get any degree three polyvectors, we would need to work in a space with more standard vectors.