Product of scalars vs vectors

• B
• smims

smims

(Scalar)·(Scalar) = Scalar
(Scalar)·(Vector) = Scalar
(Vector)·(Vector) = Scalar
(Scalar)x(Scalar) = Not valid
(Scalar)x(Vector) = Vector
(Vector)x(Vector) = Vector

Did I get them right, if not why?

Thanks

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)

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.

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.

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##.

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.

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

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.

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.