Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

B Product of scalars vs vectors

  1. May 5, 2016 #1
    (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
     
  2. jcsd
  3. May 5, 2016 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    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)
     
  4. May 5, 2016 #3

    fresh_42

    Staff: Mentor

    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. May 5, 2016 #4

    jbriggs444

    User Avatar
    Science Advisor

    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.
     
  6. May 5, 2016 #5

    fresh_42

    Staff: Mentor

    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##.
     
  7. May 5, 2016 #6

    jbriggs444

    User Avatar
    Science Advisor

    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.
     
    Last edited: May 5, 2016
  8. May 9, 2016 #7
    Thank you all for your feedback.
    The comments certainly help a lot.
     
  9. May 23, 2016 #8

    Zafa Pi

    User Avatar
    Gold Member

    It works for 7D vectors as well.
     
  10. May 23, 2016 #9

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Product of scalars vs vectors
  1. Vectors and Scalars (Replies: 3)

  2. Vector product (Replies: 5)

Loading...