Vector spaces are assumed to be finite-dimensional and over \(\bb{R}\). The grade of a multivector \(\alpha\) will be written \(\| \alpha \|\), while its magnitude will be written \(\Vert \alpha \Vert\). Bold letters like \(\b{u}\) will refer to (grade-1) vectors, while Greek letters like \(\alpha\) refer to arbitrary multivectors with grade \(\| \alpha \|\).

More notes on exterior algebra. This time, the interior product \(\alpha \cdot \beta\), with a lot more concrete intuition than you’ll see anywhere else, but still not enough.

I am not the only person who has had trouble figuring out what the interior product is for. This is what I have so far…

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1. The Interior Product

The last main tool of exterior algebra is the interior product, written \(\alpha \cdot \beta\) or \(\iota_{\alpha} \beta\). It subtracts grades (\(\| \alpha \cdot \beta \| = \| \beta \| - \| \alpha \|\)) and, conceptually, does something akin to ‘dividing \(\alpha\) out of \(\beta\)’. It’s also called the ‘contraction’ or ‘insertion’ operator. We use the same symbol as the inner product because we think of it as a generalization of the inner product: when \(\| \alpha \| = \| \beta \|\), then \(\alpha \cdot \beta = \beta \cdot \alpha = \< \alpha, \beta \>\).

Its abstract definition is that it is adjoint to the wedge product with respect to the inner product:

\[\boxed{\< \gamma \^ \alpha, \beta \> = \< \alpha, \gamma \cdot \beta \>} \tag{1}\]

In practice this means that it sort of ‘undoes’ wedge products, as we will see.

When we looked at the inner product we had a procedure for computing \(\< \b{a \^ b}, \b{c \^ d} \>\). We switched from the \(\^\) inner product to the \(\o\) inner product, by writing both sides as tensor products, with the right side antisymmetrized using \(\text{Alt}\):¹

\[\begin{aligned} \< \b{a \^ b}, \b{c \^ d} \> &= \< \b{a \o b}, \text{Alt}(\b{c \o d}) \> \\ &= \< \b{a \o b}, \b{c \o d - d \o c} \>_{\o} \\ &= \< \b{a, c} \>_V \< \b{b, d} \>_V - \<\b{a, d} \>_V \< \b{b, c} \>_V \\ &= (\b{a \cdot c}) (\b{b \cdot d}) - (\b{a \cdot d})( \b{b \cdot c}) \end{aligned}\]

Interior products directly generalize inner products to cases where the left side has a lower grade², (which is why we use \(\cdot\) for both), and can be computed with the exact same procedure:

\[\begin{aligned} \b{a} \cdot (\b{b \^ c}) &= \b{a} \cdot (\b{b \o c - c \o b}) \\ &= (\b{a} \cdot \b{b}) \b{c} - (\b{a} \cdot \b{c}) \b{b} \end{aligned}\]

A general formula for the interior product of a vector with a multivector, which can be deduced from the above, is

\[\b{a} \cdot (\b{b} \^ \beta) = (\b{a} \cdot \b{b}) \^ \beta - \b{b} \^ (\b{a} \cdot \beta)\]

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2. Projection

The intuitive meaning of the interior product is related to projection. We can construct the projection and rejection operators of a vector onto a multivector with:

\[\beta_{\b{a}} = \text{proj}_{\b{a}} \beta = \frac{1}{\| \b{a} \|^2} \b{a} \^ (\b{a} \cdot \beta) \\ \beta_{\perp \b{a}} = \text{proj}_{\perp \b{a}} \beta = \frac{1}{\| \b{a} \|^2} \b{a} \cdot (\b{a} \^ \beta)\] \[\beta = \beta_{\b{a}} + \beta_{\perp \b{a}} = \frac{1}{\vert \b{a} \vert^2} [\b{a} \^ (\b{a} \cdot \beta) + \b{a} \cdot (\b{b} \^ \beta)]\]

To understand this, recall that the classic formula for projecting onto a unit vector is:

\[\b{b}_{\b{a}} = \b{a} (\b{a} \cdot \b{b})\]

That is, we find the scalar coordinate along \(\b{a}\), then multiply by \(\b{a}\) once again. With multivectors, \(\b{a} \cdot \beta\) is not a scalar, so we can’t just use scalar multiplication – so it makes some sense that it would be replaced with \(\^\).³

\[\b{b}_{\b{a}} = \b{a} \^ (\b{a} \cdot \b{b})\]

The classic vector rejection formula is

\[\b{b}_{\perp \b{a}} = \b{b} - \b{b}_{\b{a}} = \b{b} - \b{a} (\b{a} \cdot \b{b})\]

Using the interior product we can write this as

\[\b{b}_{\perp \b{a}} = \b{a} \cdot (\b{a} \^ \b{b}) = (\b{a \cdot a}) \b{b} - (\b{a} \cdot \b{b}) \b{a} = \b{b} - \b{b}_{\b{a}}\]

The multivector version \(\b{a} \^ \beta\) is only non-zero if \(\b{\beta}\) has a component which does not contain \(\b{a}\) – all \(\b{a}\)-ness is removed by the wedge product, leaving something like \(\b{a} \^ \beta_{\perp \b{a}}\). Then \(\b{a} \cdot \b{a} \^ \beta_{\perp \b{a}} = \beta_{\perp \b{a}}\).

The correct interpretation of \(\b{a} \cdot \beta\), then, is a lot like what it means when \(\beta = \b{b}\): it’s finding the ‘\(\b{a}\)-component’ of \(\beta\). It’s just that, when \(\beta\) is a multivector, the ‘\(\b{a}\)-coordinate’ is no longer a scalar.

For example this is the ‘\(\b{x}\)‘-component of a bivector \(\b{b \^ c}\):

\[\begin{aligned} \b{x} \cdot (\b{b \^ c}) &= b_x \b{c} - c_x \b{b} \\ &= b_x (\b{c}_x + \b{c}_{\perp x}) - c_x (\b{b}_x + \b{b}_{\perp x}) \\ &= b_x \b{c}_{\perp x} - c_x \b{b}_{\perp x} \end{aligned}\]

Note that the result doesn’t have any \(\b{x}\) factors in it.

So, we might things up like this: For a unit multivector \(\alpha\), the meaning of \(\alpha \cdot \beta\) is to find \(\beta_{\alpha}\), the ‘\(\alpha\) component’ of \(\beta\).

We can remove the stipulation that \(\alpha\) be a unit multivector, but it requires being a bit careful. To illustrate why, consider an example with just vectors. What should be the value of \(v_{5x}\)? Probably as \(\frac{1}{5} v_x\), so that \(v_x \b{x} = v_{5x} 5 \b{x}\). Likewise, we need to divide through by the magnitude of \(\alpha\), so it’s actually \(\frac{1}{\Vert \alpha \Vert^2} \iota_{\alpha} \beta = \beta_{\alpha}\).

Unfortunately the rejection formula doesn’t work if \(\alpha\) is a multivector. It’s still true that \(\alpha \cdot \beta\) gives the ‘\(\alpha\)-coordinate’ of \(\beta\), if there is one. But we can only use \(\beta_{\perp \alpha} = \beta - \frac{1}{\Vert \alpha \Vert^2} \alpha \^ (\alpha \cdot \beta)\). The problem is that there are cases where both \(\alpha \^ \beta = \alpha \cdot \beta = 0\), such as for \(\b{x \^ y}\) and \(\b{y \^ z}\).⁴

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3. More identities

We can use \(\iota\) to prove a few more vector identities. First, note that \(\star\) is just a special case of \(\iota\).⁵

\[\begin{aligned} \< \alpha \cdot \omega, \star \beta \> &= \< \omega, \alpha \^ \star \beta \> \\ &= \< \omega, \< \alpha, \beta \> \omega \> \\ &= \< \alpha , \beta \> \\ &= \< \star \alpha, \star \beta \> \end{aligned}\]

Since this holds for all \(\alpha, \beta\):

\[\star \alpha = \alpha \cdot \omega\]

(1) implies that \(\iota\) obeys many of the the same rules as \(\^\):

\[\begin{aligned} (\alpha \^ \beta) \cdot \gamma &= \beta \cdot (\alpha \cdot \gamma) \\ &= (-1)^{\| \alpha \| \| \beta \|} \alpha \cdot (\beta \cdot \gamma) \\ \end{aligned}\]

Combining these, we have a way to transform applications of \(\star\):

\[\begin{aligned} \star (\alpha \^ \beta) &= (\alpha \^ \beta) \cdot \omega \\ &= \beta \cdot (\alpha \cdot \omega) \\ &= \beta \cdot (\star \alpha) \\ &= (-1)^{\| \alpha \| \| \beta \|} \alpha \cdot (\star \beta) \end{aligned}\]

Since the cross product is \(\b{a} \times \b{b} = \star (\b{a} \^ \b{b})\):

\[\b{a} \times \b{b} = \b{b} \cdot (\star \b{a}) = - \b{a} \cdot (\star \b{b})\]

This lets us unpack cross product identities. Note that \(\star^2 = (-1)^{(1)(2)} = 1\) in \(\bb{R}^3\).

Here’s the vector triple product:

\[\begin{aligned} \b{a \times (b \times c)} &= \star(\b{a} \^ \star(\b{b \^ c})) \\ &= - \b{a} \cdot \star^2 (\b{b \^ c}) \\ &= - \b{a} \cdot (\b{b \^ c}) \\ &= (\b{a} \cdot \b{c}) \b{b} - (\b{a} \cdot \b{b}) \b{c} \end{aligned}\]

The quadruple product:

\[\begin{aligned} (\b{a \times b}) \times (\b{c \times d}) &= ((\b{a \times b}) \cdot \b{d}) \b{c} - ((\b{a \times b}) \cdot \b{c}) \b{d} \\ &= \star(\b{a \^ b \^ d}) \b{c} - \star(\b{a \^ b \^ c}) \b{d} \end{aligned}\]

The Jacobi Identity:

\[\begin{aligned} 0 &= \b{a \times (b \times c)} + \b{b \times (c \times a)} + \b{c \times (a \times b)} \\ &= -{\star( \b{a} \cdot (\b{b \^ c}) + \b{b} \cdot (\b{c \^ a}) + \b{c} \cdot (\b{a \^ b}))} \\ &= -{\star( (\b{a \cdot b} - \b{b \cdot a}) \b{c} + (\b{b \cdot c - c \cdot b}) \b{a} + (\b{c \cdot a - a \cdot c}) \b{b})} \\ &= 0 \end{aligned}\]

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My other articles about Exterior Algebra:

0. Oriented Areas and the Shoelace Formula
1. Matrices and Determinants
2. The Inner product
3. The Hodge Star
4. The Interior Product
5. EA as Linearized Set Theory?
6. Oriented Projective Geometry
7. Simplex Volumes and Boundaries
8. All the Exterior Algebra Operations