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

All below, well

## Relevant Equations:

All below

That is, the triad forms a basis.

etotheipi

## Answers and Replies

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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.

Infrared
Gold Member
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.##

LCSphysicist
Mark44
Mentor
[a,b,c] = a ^ b . c
What does this notation mean, particularly '^'?

Infrared
Gold Member
I assumed cross/exterior product (the latter is more general, but in ##\mathbb{R}^3## is basically equivalent). The latex would be "##\wedge##".

LCSphysicist
etotheipi
Gold Member
2019 Award
So does the notation ##[\vec{a}, \vec{b}, \vec{c}]## mean the vector triple product (in ##\mathbb{R}^3##)?

Infrared
Gold Member
It means the scalar triple product, ##[a,b,c]=(a\times b)\cdot c.##

LCSphysicist and etotheipi
etotheipi
Gold Member
2019 Award
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!