Cross product or vector product.

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Discussion Overview

The discussion revolves around the interpretation of the angle theta in the context of the cross product of vectors. Participants explore its geometric meaning, particularly when vectors are not positioned tail to tail, and the implications for determining the resulting vector perpendicular to both original vectors. The scope includes conceptual clarification and technical reasoning related to vector mathematics.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question the definition of theta as the angle at which two vectors coincide in direction, seeking further clarification on its meaning.
  • Others suggest that when vectors are not connected tail to tail, they should be adjusted to meet at their tails to properly define the angle between them.
  • A participant emphasizes that the angle is determined by the directions of the vectors and that there is no concept of a "smallest" angle in this context.
  • Another point raised is that the primary goal of the cross product is to find the signed area of the parallelogram formed by the vectors, which involves measuring the angle from the first vector to the second.
  • There is a reiteration that in three dimensions, a nonzero area indicates a unique line perpendicular to the parallelogram formed by the vectors.

Areas of Agreement / Disagreement

Participants express differing views on the definition and significance of the angle theta in the cross product, with no clear consensus reached on the best approach to understanding it.

Contextual Notes

Some statements rely on specific interpretations of vector positioning and angle measurement, which may not be universally agreed upon. The discussion reflects varying levels of understanding regarding the geometric implications of the cross product.

prashant singh
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What does the angle theta acutally means in cross product because I have seen in many places it is written that theta is the angle at which two vector on a given plane will coinside with each other so that there will be only one direction. Is it true and why they defined it in this way , I need more information on this
 
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prashant singh said:
What does the angle theta acutally means in cross product because I have seen in many places it is written that theta is the angle at which two vector on a given plane will coinside with each other so that there will be only one direction. Is it true and why they defined it in this way , I need more information on this
Have you read this: https://en.wikipedia.org/wiki/Cross_product#Geometric_meaning ?
 
If there are two vector in a given plane and they are not connected tail to tail then what we have to doo. I think now we have to connect the two vectors tail to tail by the smallest angle theta so that we can have a vector perpendicular to both the vectors. Correct me if I am wrong
fresh_42 said:
Have you read this: https://en.wikipedia.org/wiki/Cross_product#Geometric_meaning ?
 
prashant singh said:
If there are two vector in a given plane and they are not connected tail to tail then what we have to doo. I think now we have to connect the two vectors tail to tail by the smallest angle theta so that we can have a vector perpendicular to both the vectors. Correct me if I am wrong
A vector is a direction and a length. Where you "connect" them depends on what you want to do.
E.g. a force has a direction and an amount. Whether you can use it to pull the chair depends on where you apply the force.
I think you're on the right track. Cancel the word "smallest" and it looks ok. The angle is defined by the directions the vectors point. There is no smallest or biggest. Of course you have to put them tail to tail to determine the angle between them.
 
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In the cross-product, the first goal should be to determine the signed-area of the parallelogram formed by the vectors.
Slide the vectors to have their tails meet, then measure the angle from the first vector to the second vector.
Then you'll take the sine of that angle.
In three dimensions, if this area is nonzero,
there will be a unique line that is perpendicular to that parallelogram.
 
Thanks brooo , cooool
robphy said:
In the cross-product, the first goal should be to determine the signed-area of the parallelogram formed by the vectors.
Slide the vectors to have their tails meet, then measure the angle from the first vector to the second vector.
Then you'll take the sine of that angle.
In three dimensions, if this area is nonzero,
there will be a unique line that is perpendicular to that parallelogram.
 

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