Why does (a.b).c make no sense?

  1. I was studying the dot product, and it says that (a.b).c makes no sense.

    so if you do (a.b) can = to β
    and then is it not possible to do β.c?

    WHY cant you 'dot' a scalar and a vector? why?
     
  2. jcsd
  3. Mark44

    Staff: Mentor

    Because the dot product is defined ONLY for two vectors. You can multiply a vector by a scalar, and this is called scalar multiplication.
     
    1 person likes this.
  4. But why? is there any proofs as to why this is defined this way?
     
  5. Have you tried it? Write down the definition of a dot product. Make up and write down three vectors and perform the calculation.

    (Note that definitions are made up, not proved. Can you prove that a cat is not a soda can? No. Its just not defined that way. Theorems and identities are what get proved, under the right definitions.)
     
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  6. Matterwave

    Matterwave 3,852
    Science Advisor
    Gold Member

    The dot product, in two dimensions (for simplicity) is defined as:

    $$\vec{a}\cdot \vec{b}=a_xb_x+a_yb_y$$

    Now, this assumes ##\vec{a}=(a_x,a_y)## and ##\vec{b}=(b_x,b_y)## are vectors. What would it mean to turn ##a## into a number? Certainly you can "define" the "dot product" of a scalar and a vector as:

    $$a\cdot\vec{b}=a\vec{b}=(ab_x,ab_y)$$

    But that's just the same as a scalar product, so it would be supremely confusing to also call it a "dot product". That's why we don't call that the "dot product".
     
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  7. thank you!
     
  8. Matterwave

    Matterwave 3,852
    Science Advisor
    Gold Member

    No problem. =]
     
  9. Mark44

    Staff: Mentor

    A definition doesn't have to be proved.
     
  10. What an unexpected place to find such a gem.
     
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