Cross product or vector product.

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SUMMARY

The discussion centers on the geometric interpretation of the angle theta in the context of the cross product of two vectors. Participants clarify that theta represents the angle between two vectors when positioned tail-to-tail, which is essential for calculating the sine of that angle. The cross product yields a vector that is perpendicular to the plane formed by the two original vectors, with the magnitude corresponding to the signed area of the parallelogram defined by them. The conversation emphasizes the importance of correctly positioning the vectors to accurately determine this angle.

PREREQUISITES
  • Understanding of vector operations, specifically cross product
  • Familiarity with geometric interpretations of vectors
  • Knowledge of sine function and its application in trigonometry
  • Basic concepts of three-dimensional space and vector orientation
NEXT STEPS
  • Study the geometric meaning of the cross product in detail
  • Learn how to calculate the sine of an angle between two vectors
  • Explore the properties of the parallelogram formed by two vectors
  • Investigate applications of the cross product in physics, such as torque and angular momentum
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Students and professionals in mathematics, physics, and engineering who seek to deepen their understanding of vector mathematics and its applications in three-dimensional space.

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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