Excuse me ... .

A vector or polar-vector can denote as in a tangent space on point at an -dimensional manifold , where is a basis in ... .

The dot product is a component of metric tensor on the manifold ... .

For each , the dot product ,

is a scalar,

and the cross product is a pseudo-vector alias axial-vector, as I have known ... .

For each , the triple scalar product is defined as which totally antisymmetric by a permutation ... . is a pseudo-scalar, as I have known ... ,

and the triple vector product is a vector alias polar-vector ... .

The other combinations are

• is a scalar ... , and

• is a pseudo-vector alias axial-vector, as I have known ... ,

for each ... .

 U V U•V U×V [vector] [vector] [scalar] [pseudo-vector] [vector] [pseudo-vector] [pseudo-scalar] [vector] [pseudo-vector] [vector] [pseudo-scalar] [vector] [pseudo-vector] [pseudo-vector] [scalar] [pseudo-vector]

What do multiplication scalar with scalar, scalar with pseudo-scalar, pseudo-scalar with pseudo-scalar, scalar with vector, scalar with pseudo-vector, pseudo-scalar with vector, and pseudo-scalar with pseudo-vector yield ... ?

 u v uv [scalar] [scalar] [scalar] [scalar] [pseudo-scalar] [pseudo-scalar] [pseudo-scalar] [pseudo-scalar] [???]

 u V uV [scalar] [vector] [vector] [scalar] [pseudo-vector] [pseudo-vector] [pseudo-scalar] [vector] [pseudo-vector] [pseudo-scalar] [pseudo-vector] [???]

What are the two [???] ‘s in the last two tables ... ?

Are the last two tables right ... ?

Thank you very much for the answer ... . 