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Today we were studying vector and scalar products, when my teacher gave an example of them and told us that area is a vector quantity.

I Have studied previous class’s physics books (two years ago) which defined vector as any quantity that has both direction and magnitude while scalar is defined as any quantity that has only has magnitude and not direction.

And if I remember my high elementary mathematics then:

Area is equal to length x width. Both of these quantities have SI units meter. So we have meter as length, then knowing that length is a scalar quantity:

Scalar quantity x scalar quantity = scalar quantity

Example :

Speed: $$ Speed = v = \frac{d}{t}$$ Where $d$ is distance and $t$ time, which are both scalar quantities but knowing

Unit is : m/s

Velocity:

$$Velocity = \mathbf v = \frac{\mathbf D}{t}$$

where $\mathbf D$ is displacement (a vector) and $t$ the time (a scalar).

Unit is : m/s.

So the question is why did my teacher say that area, a product of two scalar quantities, is vector in nature?

Kyle Kanos
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Muhammad Salman
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  • The clearest explanation of this that I know of is in Banesh Hoffmann's About Vectors. We can explain it here but there's no substitute for a full textbook walkthrough. – Emilio Pisanty May 23 '18 at 12:00
  • @EmilioPisanty : while you are right on no substitute for a full text book. Reading this will take time. – Muhammad Salman May 23 '18 at 12:04
  • This is a common problem when people first learn physics: the issue is that you can't learn everything from dimensional analysis alone. – knzhou May 23 '18 at 12:14
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    An area can be associated with a direction: it is the direction perpendicular to the area. You can't have figured that out by dimensional analysis alone, because this only works in three dimensions: in two dimensions there is no perpendicular direction and in four dimensions there is more than one. – knzhou May 23 '18 at 12:15
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    Possible duplicate of How can area be a vector? – Kyle Kanos May 23 '18 at 12:18
  • @KyleKanos : I cant make head or tails of what you linked :( – Muhammad Salman May 23 '18 at 12:21
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    The mathematical definition of a vector is more comprehensive than the introductory model provided by an introductory physics class; you will find that there is more to learn even after you take a full linear algebra class! In any case, a vector is an element of a vector space. When viewed against the definitions, the real numbers form a vector space, as does the space of polynomials, etc. etc. The nomenclature of "scalar, vector, tensor" derives from a scheme where tensors are used, and one is seeking the invariants. – Peter Diehr May 23 '18 at 12:32
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    @MuhammadSalman Then you should edit your question to say explicitly which aspects of the linked duplicate's answers you don't understand. As it currently stands, your question is quite likely to be closed as a duplicate. – Emilio Pisanty May 23 '18 at 12:33
  • @EmilioPisanty : Ok. – Muhammad Salman May 23 '18 at 12:40

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