Using 3 Vectors to Show Vector Multiplication is Not Commutative

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SUMMARY

This discussion focuses on demonstrating that vector multiplication is not commutative using three specific vectors in 3-dimensional space. Participants emphasize the importance of calculating the cross products correctly, specifically a × b and b × c, and then applying the results to find (a × b) × c and a × (b × c). The necessity of verifying that the resulting vectors are perpendicular through the dot product is also highlighted as a crucial step in the solution process.

PREREQUISITES
  • Understanding of vector operations, specifically cross product
  • Familiarity with 3-dimensional vector space concepts
  • Knowledge of the dot product and its properties
  • Ability to interpret and manipulate mathematical notation in vector calculus
NEXT STEPS
  • Study the properties of the cross product in vector algebra
  • Learn how to calculate and interpret the dot product of vectors
  • Explore the geometric interpretation of vector multiplication
  • Review examples of non-commutative operations in linear algebra
USEFUL FOR

Students studying vector calculus, educators teaching linear algebra concepts, and anyone interested in understanding the properties of vector multiplication in physics and mathematics.

amy098yay
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Homework Statement


That is, use three specific vectors in 3-space to show that https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117×(https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-c.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) is not equal to (https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-c.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117.

The Attempt at a Solution


the solution is in the pdf file, did i make a mistake in answering the question..?
 

Attachments

Physics news on Phys.org
..
 
amy098yay said:

Homework Statement


That is, use three specific vectors in 3-space to show that https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117×(https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-c.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) is not equal to (https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-a.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117 × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-b.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117) × https://ucdsb.elearningontario.ca/content/enforced/4850117-BL_1415Sem2__MAT_MCV4UU-948314_1_ELO/MCV4UPU01/MCV4UPU01A06/images/vec-c.gif?_&d2lSessionVal=Y3hirJUTSYjH76OEZwqHIBATE&ou=4850117.

The Attempt at a Solution


the solution is in the pdf file, did i make a mistake in answering the question..?
It's hard to follow your work, so I didn't check it. For the first triple product, please show us how you did b X c, and then a X (b X c). For the second triple product, please show is a X b, and then (a X b) X c.

As a self-check for your work, you should verify that when you calculate a X b, for example, the vector you get is perpendicular to both a and b. This can be done very quickly using the dot product - the dot product of perpendicular vectors is 0.
 
another pdf file of the solution
 

Attachments

amy098yay said:
another pdf file of the solution

axb and bxc are ok. I have no idea what you are doing when you try to find (axb)xc and ax(bxc).
 
This is the same sort of problem you're having in the other thread, https://www.physicsforums.com/threads/vectors-need-help.800394/.
 

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