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

[LCSphysicist]

Homework Statement

All below, well

Relevant Equations

All below

That is, the triad forms a basis.

 

[LCSphysicist]

[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]

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 $e_3.$

 

[Mark44]

[a,b,c] = a ^ b . c

What does this notation mean, particularly '^'?

 

[Infrared]

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

 

[etotheipi]

So does the notation $[\vec{a}, \vec{b}, \vec{c}]$ mean the vector triple product (in $\mathbb{R}^3$)?

 

[Infrared]

It means the scalar triple product, $[a,b,c]=(a\times b)\cdot c.$

 

[etotheipi]

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!