Understanding Scalar and Vector Products in Geometric Algebra

[smims]

(Scalar)·(Scalar) = Scalar
(Scalar)·(Vector) = Scalar
(Vector)·(Vector) = Scalar
(Scalar)x(Scalar) = Not valid
(Scalar)x(Vector) = Vector
(Vector)x(Vector) = VectorDid I get them right, if not why?

Thanks

 

[stevendaryl]

By "x" do you mean the vector cross-product? And by "." do you mean the vector scalar product? If the answer is yes, then those operations are only applicable to a pair of vectors. I would say there are four common types of "multiplication" involving scalars and 3D vectors:

1. Scalar times scalar to produce a scalar (ordinary multiplication)
2. Scalar times vector to produce a vector (scaling a vector)
3. Vector times vector to produce a scalar (scalar or "dot" product)
4. Vector times vector to produce a vector ("cross" product)

 

[fresh_42]

5. Scalar cross scalar to denote the pair (scalar,scalar). "Not valid" is misleading here.
6. Scalar cross vector also denotes a pair and no vector. If it is supposed to be a vector, the cross has to be explicitly defined, e.g as scaling.

 

[jbriggs444]

5. Scalar cross scalar to denote the pair (scalar,scalar). "Not valid" is misleading here.

You are interpreting AxB for scalars A and B in the sense of the Cartesian product of a pair of sets? [the set of ordered pairs (a,b) where a is an element of A and b is an element of B]. That is rather a stretch since A and B are not sets and their cross product would not be an ordered pair but would, rather, be a [possibly singleton] set of ordered pairs.

 

[fresh_42]

You are interpreting AxB for scalars A and B in the sense of the Cartesian product of a pair of sets? [the set of ordered pairs (a,b) where a is an element of A and b is an element of B]. That is rather a stretch since A and B are not sets and their cross product would not be an ordered pair but would, rather, be a [possibly singleton] set of ordered pairs.

Agreed. I simply found "not valid" a bit harsh, esp. because the word scalar has been put into brackets. Sometimes I've also read lines like $2 \times 2 = 4$.

 

[jbriggs444]

Agreed. I simply found "not valid" a bit harsh, esp. because the word scalar has been put into brackets. Sometimes I've also read lines like $2 \times 2 = 4$.

If one insists on intepreting $\times$ in the sense of the Cartesian product and interpreting 2 in the sense of set theory and the Von Neumann construction then it would be more correct to say that $|2 \times 2| = 4$ and that $2 \times 2$ = { (0,0), (0,1), (1,0), (1,1) }

This follows since, in the Von Neumann construction, 2 = {0,1}.

Bringing us back on topic for this thread... One should interpret the $\times$ notation according to context. In the context of $2 \times 2$ and a discussion of scalars and vectors, it cannot reasonably denote either a cross product of vectors or a Cartesian product of sets. The most reasonable interpretation would as an ordinary product of integers.

 

[smims]

Thank you all for your feedback.
The comments certainly help a lot.

 

[Zafa Pi]

By "x" do you mean the vector cross-product? And by "." do you mean the vector scalar product? If the answer is yes, then those operations are only applicable to a pair of vectors. I would say there are four common types of "multiplication" involving scalars and 3D vectors:

1. Scalar times scalar to produce a scalar (ordinary multiplication)
2. Scalar times vector to produce a vector (scaling a vector)
3. Vector times vector to produce a scalar (scalar or "dot" product)
4. Vector times vector to produce a vector ("cross" product)

It works for 7D vectors as well.

 

[micromass]

It works for 7D vectors as well.

It works in any dimension with geometric algebra as well. Also, the 7D cross product is unsatisfactory: no Jacobi identify and it's not unique.