Tensor products of linear combinations are not expanded automatically:

Expand the product:

Expand a scalar product of two cross products:

Act on expressions containing all kinds of products:

Act on expressions containing both symbolic and explicit objects:

For some operations, additional information needs to be supplied. The Dot product is not commutative in general:

However, if the objects are declared as vectors, then it is commutative:

Linearity expansions:

Vector identities in dimension 3:

Vector identities in dimension 4:

Combination of transpositions:

Powers of matrices. Matrices are assumed invertible:

Declare a, b, c to be three-dimensional vectors:

Then these formulas effectively invert the matrix {a,b,c}:

Expansion of pairs of vector products requires consistent dimensions:

With symbolic expressions, the result may be given as 0, representing a 0 vector or tensor:

Compare with:

Declare a family of vectors:

For each integer n, there is a tree expression involving n Cross operations for n+1 vectors v₀,…,v_n:

Now systematically convert all possible combinations of cross products into dot products and one cross product: