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We will need the cross product of two vectors a few times, so let us begin with a definition for it:

cross (v1, v2) := determinant(matrix([[1, 0, 0], [0, 1, 0], [0, 0,1]], v1, v2));

Here is an example:

 cross([12, -1, 3], [-9, -2, -2]);
  [8, - 3, - 33]

From:Maxima tutorial 

2. how to compute dot product?

Just using the dot operator to compute it.

Example:

v1.v2;

3. VECTORS

- The file SHARE;VECT > contains a vector analysis package, share/vect.dem contains a corresponding 
demonstration, and SHARE;VECT ORTH contains definitions of various orthogonal curvilinear coordinate
 systems. LOAD(VECT); will load this package for you. The vector analysis package can combine and 
simplify symbolic expressions including dot products and cross products, together with the gradient, 
divergence, curl, and Laplacian operators. The distribution of these operators over sums or products
 is under user control, as are various other expansions, including expansion into components in any 
specific orthogonal coordinate systems. There is also a capability for deriving the scalar or vector
 potential of a field. The package contains the following commands: VECTORSIMP, SCALEFACTORS, EXPRESS,
 POTENTIAL, and VECTORPOTENTIAL. Do DESCRIBE(cmd) on these command names, or PRINTFILE(VECT,USAGE,SHARE);
 for details. Warning: The VECT package declares "." to be a commutative operator.