Wave Vector Algebra: Proving k = 2∏/λ

[Syrus]

Homework Statement

I'm having difficulty showing that the general equation k = 2∏/λ holds from the component equations k_x = 2∏/λ_x etc..., k = √(k_x² + k_y² + k_z²), and λ = √(λ_x² + λ_y² + λ_z²). Any help? There is a photo from a textbook with the equations also below.

Homework Equations

The Attempt at a Solution

 

[BruceW]

what are you defining λ to be?

edit: also, if you use λ in the equation k = 2∏/λ then you can see it doesn't work.

 

[Syrus]

Yes, I'm just tryin to derive the formula for λ from the component formulae, but obviously (as you state) it doesn't seem to work. What then, is the justification for the association between the general wavenumber and the component wavenumbers if they don't correspond in the natural way?

 

[BruceW]

the general wavenumber does correspond in the natural way to the component wavenumbers. But the general wavenumber does not correspond in the natural way to the component wavelengths. Is this what you were thinking about?

 

[Syrus]

Yes, precisely. I wonder, then, how the component wavenumbers (and their corresponding component wavelengths) are related to the general wavenumber and wavelength?

 

[BruceW]

if you think about the vector wavenumber and vector wavelength: (kx,ky,kz) (call it k) and (λx,λy,λz) (call it λ) then there is a nice equation you can write, which involves the two vectors k and λ (hint: what kind of operation can you use between two vectors?)