Prove that the components of a vector can be written as follows

  • #1
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Homework Statement:

All below, well

Relevant Equations:

All below
1592759390018.png

That is, the triad forms a basis.
 
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Answers and Replies

  • #2
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[a,b,c] = a ^ b . c

I thought that i could begin with a matrix, immediately i see the problem seeing by this way, so "never mind". I don't know how the properties would help me here, i think i will need to evaluate by geometry approach, but i am not sure.
 
  • #3
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Since ##\{e_1,e_2,e_3\}## is a basis, you can uniquely write ##x=a_1e_1+a_2e_2+a_3e_3.## To solve for ##a_1##, for example, try crossing both sides with ##e_2## and then dotting both sides by ##e_3.##
 
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  • #5
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I assumed cross/exterior product (the latter is more general, but in ##\mathbb{R}^3## is basically equivalent). The latex would be "##\wedge##".
 
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  • #6
etotheipi
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So does the notation ##[\vec{a}, \vec{b}, \vec{c}]## mean the vector triple product (in ##\mathbb{R}^3##)?
 
  • #7
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It means the scalar triple product, ##[a,b,c]=(a\times b)\cdot c.##
 
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  • #8
etotheipi
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It means the scalar triple product, ##[a,b,c]=(a\times b)\cdot c.##
Whoops, yes that's the one I was thinking of. Apparently what's called the vector triple product is something else. Makes sense, thanks!
 

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