For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of
- the interior (or scalar) product $\mathbf u\cdot \mathbf v$,
- the cross product $\mathbf u\times \mathbf v$ (if it exist; I'm not sure it's defined for general multivectors), and
- the exterior (or wedge) product $\mathbf u\wedge \mathbf v$, respectively?
The closest thing to general definitions I have found for these products are on the Wikipedia page for bivectors, where they are defined as follows:
- The interior product between two (simple) vectors $\mathbf a$ and $\mathbf b$ is defined as $\mathbf a\cdot \mathbf b=\frac{1}{2}(\mathbf a\mathbf b + \mathbf b\mathbf a)$
- The cross product between two bivectors $\mathbf A$ and $\mathbf B$ is defined as $\mathbf A\times \mathbf B=\frac{1}{2}(\mathbf A\mathbf B - \mathbf B\mathbf A)$
- The exterior product between two vectors $\mathbf a$ and $\mathbf b$ is defined as $\mathbf a\wedge \mathbf b=\frac{1}{2}(\mathbf a\mathbf b - \mathbf b\mathbf a)$
However, this is a bit unsatisfactory, because it doesn't define the products for general multivectors, but only for simple vectors or bivectors. Note that the definitions for the cross product and for the exterior product are almost the same, with the only difference being that they concern vectors of different grades, so I assume that at least one of those definitions doesn't hold for general multivectors, since otherwise they would be exactly the same.
$ \newcommand\form[1]{\langle#1\rangle} \newcommand\G{\mathscr G} \newcommand\lcontr{\mathbin\rfloor} \newcommand\rcontr{\mathbin\lfloor} $Any decent book on geometric algebra should have these formulas. See for instance Geometric Algebra for Physicists by Doran and Lasenby or Clifford Algebras and Spinors by Pertti Lounesto.
Define $\form\cdot_k$ as the projection onto $k$-vectors, with the special case that $k<0$ always yields $0$.
Write $\G^k$ for the space of $k$-vectors. The whole algebra $\G$ is of the form $$ \G = \G^0\oplus\G^1\oplus\G^2\oplus\dotsc\oplus\G^n $$ where $\G^0 = \mathbb K$ is our field of scalars, $\G^1 = V$ is our underlying vector space, and $n = \dim V$. So it enough to define any multilinear operation on homogeneous multivectors and then extend by linearity.
Write $A, B, C$ for arbitrary multivectors and $A_s, B_t$ for arbitrary $s$-vectors and $t$-vectors, respectively.
The scalar product is $$ A\cdot B = \form{AB}_0. $$
The "interior product", more properly known as the contraction, comes in left and right variants: $$ A_s\lcontr B_t = \form{A_sB_t}_{t-s},\quad A_s\rcontr B_t = \form{A_sB_t}_{s-t}. $$ Some authors (in particular Doran and Lasenby) define a symmetric "inner product" using $|s-t|$, but the contractions have much better algebraic properties. For instance, they can also be defined as the adjoints of the exterior product under the scalar product: $$ A\cdot(B\wedge C) = (A\rcontr B)\cdot C,\quad (A\wedge B)\cdot C = A\cdot(B\lcontr C). $$ If $A_1$ is a vector then $$ A_1\lcontr B = \frac12(A_1B - \hat BA_1) $$ where the grade involution $B\mapsto \hat B$ is the unique algebra automorphism that negates vectors. We have a similar equation for the right contraction. It's possible to extend this into a definition for arbitrary $A$, but that process is too involved to get into here.
The symbol $\times$ is not used to mean "cross product" in geometric algebra; instead it is usually just the commutator product $$ A\times B = \frac12(AB - BA). $$ This is particularly useful for bivectors because $A_2\times B_t$ has grade $t$; this also means we can write $$ A_2\times B_t = \form{A_2B_t}_t. $$
The exterior product is $$ A_s\wedge B_t = \form{A_sB_t}_{s+t}. $$ For a vector $A_1$ it has a formula like the contractions do: $$ A_1\wedge B = \frac12(A_1B + \hat BA_1). $$ Similarly, this can also be extended to a definition for arbitrary $A$. Putting both together gives $$ A_1B = A_1\lcontr B + A_1\wedge B. $$ We have a similar formula for bivectors: $$ A_1B = A_1\lcontr B + A_1\times B + A_1\wedge B. $$ These two formulas also follow simply from the fact that $$ A_sB_t = \form{A_sA_t}_{|s-t|} + \form{A_sA_t}_{|s-t|+2} + \form{A_sA_t}_{|s-t|+4} + \dotsb + \form{A_sA_t}_{s+t}. $$