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

Definition of vector addition, Cartesian product?

  1. Jun 23, 2012 #1
    I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V [itex]\times[/itex] V [itex]\rightarrow[/itex] V. My text describes V as a set and describes the above process as a mapping. I believe the [itex]\times[/itex] may represent a Cartesian product. Could someone fill me in on how such an operation could define vector addition?
  2. jcsd
  3. Jun 23, 2012 #2


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Vector addition just takes two vectors and gives you third. So the map is

    (X,Y) -> X + Y.
  4. Jun 23, 2012 #3
    It's a product of sets, which we map out of with addition. Yes, it is the Cartesian product, in the sense that Cartesian product is product of sets. You can map any point in the product (x,y), as lavinia said, to x+y for instance.

    You might contrast product and/or sum of sets with product and/or sum of elements.
  5. Jun 23, 2012 #4


    User Avatar
    Science Advisor
    Gold Member

    You could do the same for normal addition (i.e. the addition of the real numbers).

    I.e. you define a function +:ℝ2→ℝ (where ℝ2=ℝ×ℝ)

    For example +(13,2) = 15

    Infact +:ℝ2→ℝ is actually an example of vector addition as the reals themselves form a 1-D real vector space wrt real addition and real multiplication.
    Last edited by a moderator: Jun 24, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook