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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsVector AlgebraProperties of Cross Product

Cross Product : Not Commutative

In this page, you will learn that the vector cross product is not commutative.



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What does 'commute' mean?commutative property illustration

  • to go to and fro on a regular basis
  • to go to and fro on a regular basis
  • it does not mean anything

The answer is 'to go to and fro between two places on a regular basis'.

Given the definition of cross product as
`vec p xx vec q = |vec p||vec q|sin theta hat n`
What is `vec q xx vec p`?

  • `vec p xx vec q`
  • `|p||q|sin(theta) hat n`
  • `|p||q|sin(-theta) hat n`
  • `|p||q|sin(-theta) hat n`
  • all the above

The answer is '`|p||q|sin(-theta) hat n`'. The angle traversed from `vec q` to `vec p` is `-theta`.

Given the definition of cross product as
`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`
What is `vec q xx vec p`?

  • `-vec p xx vec q`
  • `|(i, j, k),(q_x, q_y, q_z),(p_x, p_y, p_z)|`
  • `-|(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`
  • all the above
  • all the above

The answer is 'All the above'

 •  In a cross product if the vectors are swapped, the result is negative of the product. - not commutative.

Not Commutative: For any vector `vec p, vec q in bbb V`
`vec p xx vec q = - vec q xx vec p`

Solved Exercise Problem:

Given that `vec p xx vec q = 2.1i-1.7j+6k`, find the `vec q xx vec p`.

  • `-2.1i+1.7j-6k`
  • `-2.1i+1.7j-6k`
  • `2.1i-1.7j+6k`
  • `sqrt(2.1^2+1.7^2+6^2)`
  • `2.1^2+1.7^2+6^2`

The answer is '`-2.1i+1.7j-6k`'.

                            
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