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

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Discussion Overview

The discussion revolves around the nature of the dot product in vector mathematics, specifically addressing why the expression (a.b).c is considered nonsensical. Participants explore the definitions and implications of dot products and scalar multiplication, questioning the foundational aspects of these operations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that the dot product is defined only for two vectors, which leads to the conclusion that (a.b).c cannot be computed.
  • Others propose that while (a.b) can be treated as a scalar (β), it does not allow for a dot product with a vector (c) because a scalar cannot be dotted with a vector.
  • A participant questions the reasoning behind the definition of the dot product, seeking proofs or justifications for why it is defined in this manner.
  • Another participant suggests that definitions are not necessarily proven, using the analogy of proving that a cat is not a soda can to illustrate this point.
  • One participant provides a mathematical definition of the dot product in two dimensions, explaining how it operates under the assumption that both operands are vectors.
  • There is a suggestion that defining the dot product of a scalar and a vector would lead to confusion, as it would overlap with the concept of scalar multiplication.

Areas of Agreement / Disagreement

Participants express differing views on the nature of definitions in mathematics, with some emphasizing the lack of necessity for proofs, while others seek deeper understanding. The discussion remains unresolved regarding the foundational reasoning behind the definitions of dot products and scalar multiplication.

Contextual Notes

Participants highlight that definitions in mathematics can be arbitrary and not subject to proof, which introduces a layer of complexity in understanding why certain operations are defined as they are.

uzman1243
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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 can't you 'dot' a scalar and a vector? why?
 
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uzman1243 said:
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 can't you 'dot' a scalar and a vector? why?

Because the dot product is defined ONLY for two vectors. You can multiply a vector by a scalar, and this is called scalar multiplication.
 
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Mark44 said:
Because the dot product is defined ONLY for two vectors. You can multiply a vector by a scalar, and this is called scalar multiplication.

But why? is there any proofs as to why this is defined this way?
 
uzman1243 said:
But why? is there any proofs as to why this is defined this way?

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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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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Matterwave said:
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".

thank you!
 
No problem. =]
 
uzman1243 said:
But why? is there any proofs as to why this is defined this way?
A definition doesn't have to be proved.
 
ModusPwnd said:
Can you prove that a cat is not a soda can?

What an unexpected place to find such a gem.
 

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