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While I was reading dot and cross products I stumbled upon the well known fact that the dot product yields a scalar and the cross product yields a vector. All I want to know is why. Why does the dot and cross products always yields Scalars and vectors respectively. And also, what led us to define the products the way they are defined?

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    $\begingroup$ I suppose thew come from Physics, where some quantities arise in a natural way as a dot product (e.g. work) or a vector product (e.g. torque). $\endgroup$ Oct 1, 2023 at 17:08
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    $\begingroup$ See also quaternions, a system that preceded vectors. There we find "inner product" and "outer product" which later became dot product and cross product. en.wikipedia.org/wiki/Quaternion $\endgroup$
    – GEdgar
    Oct 1, 2023 at 17:12

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I think this question is a bit backwards. Asking "why does the dot product always yield a scalar?" is like asking "why is red a color?"

There's no possible answer to this question: red is a color. "Red" is a word which we use to refer to a color, so of course red is a color. There's no reason for it; it's just part of what "red" means.

Likewise, there's no answer to the question "why does the dot product always yield a scalar?". "Dot product" is just the phrase we use to refer to a particular operation which always yields a scalar.


The more important question is "why do we care about these operations of dot and cross product in particular?". The answer is that they are useful, because we can easily compute these operations and learn things about vectors from the computation. For example, two vectors are orthogonal if and only if their dot product is $0$. This gives you a reasonable way to check if two vectors in 700-dimensional space are orthogonal, for example. It's not like you can pull out a protractor in 700-dimensional space and measure angles directly, so this is a very useful computational tool to have!

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  • $\begingroup$ Ok. Now what about the definition part? How do we know that we'll need the cosine of the angle between the vectors to obtain a scalar, why not sine? Your answers will be greatly appreciated. $\endgroup$ Oct 2, 2023 at 13:50
  • $\begingroup$ Well, the sine of an angle is also a scalar of course. We could define a new operation which tells us the sine of the angle between two vectors (* their magnitudes). But this would be less useful for two reasons: (1) It would have a more complicated formula than the one for the dot product (2) The cosine of the angle between two vectors can be positive or negative, and the sign tells you something important (is the angle acute or obtuse?). On the other hand, the sine of the angle between two vectors is always positive -- you can't tell if an angle is acute or obtuse if you only know its sine. $\endgroup$ Oct 2, 2023 at 14:14
  • $\begingroup$ In math you can make absolutely any definition you like, as long as it makes sense. You are not ever limited to the concepts you learn in courses! The ideas and definitions and techniques you'll learn about in math courses are just the ones that have proven to be most useful and interesting over hundreds of years of study. $\endgroup$ Oct 2, 2023 at 14:16
  • $\begingroup$ I greatly appreciate your help sir. Thank you for your kind attention! It's now clear. $\endgroup$ Oct 2, 2023 at 14:17

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