About vectors and vector products

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Discussion Overview

The discussion revolves around the nature of products involving physical quantities, specifically focusing on when the result of such multiplications will be a scalar or a vector. Participants explore the distinctions between different types of vector products, namely the dot product and the cross product, and the conditions under which each is applicable.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes that understanding the result of multiplying two physical quantities requires context, as the nature of the quantities can vary widely (e.g., scalars, vectors, tensors).
  • Another participant emphasizes that the terms "being multiplied" lacks specificity, as it could refer to various operations depending on the context.
  • A claim is made that the dot product and cross product are mathematically defined operations, resulting in a scalar and a vector, respectively, with physics providing specific interpretations of these results.
  • It is mentioned that the dot product can yield the angle between two vectors, while the cross product produces a new vector that may represent physical properties such as angular momentum or torque.
  • The geometric interpretation of the cross product is discussed, specifically that its magnitude corresponds to the area of a parallelogram defined by the two vectors involved.

Areas of Agreement / Disagreement

Participants express varying levels of agreement on the definitions and implications of vector products, but there is no consensus on a unified approach to determining the nature of the result when multiplying physical quantities.

Contextual Notes

The discussion highlights the ambiguity in the terms used, such as "physical quantities" and "being multiplied," which may depend on specific definitions and contexts that are not fully explored.

kapoor_kapoor
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If two physical quantities are being multiplied , is there any way to know that the result will be a scalar of vector.. Moreover if two vectors are being multiplied how can we know that we have to apply cross or dot product??
 
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You need some context. "Two physical quantities" is too vague for it to be possible to know what is going on. Physical quantities can be many different forms. Temperature would presumably be a scalar. The velocity field of a fluid would be a vector field. The stress in a solid is probably expressed as a tensor. And so on.

"Being multiplied" is likewise too vague to understand what is going on. For example, some quantities are a scalar times a scalar in some contexts. In other contexts they might be a scalar times a vector. Or a cross product. Or a dot product.
 
Dot product and cross product are mathematically defined vector operations, resulting in a scalar or a vector respectively. Physics gives particular meaning to the results.
 
kapoor_kapoor said:
If two physical quantities are being multiplied , is there any way to know that the result will be a scalar of vector.. Moreover if two vectors are being multiplied how can we know that we have to apply cross or dot product??

A dot product will get you the angle between two vectors.

A cross product will get you a new vector that presumably represents some physical property (angular momentum, torque, etc). Geometrically, the magnitude of the cross product gives you the area of a parallelogram whose sides are defined by the two vectors you took the cross product of.
 
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