Plane EM wave in a vacuum, quick identity question

[binbagsss]

Okay the question is, given a plane electromagnetic wave in a vacuum given by E=(Ex,Ey,Ez)exp$^{(i(k_{x}x+k_{y}y+k_{z}z-wt)}$ and B=(Bx,By,Bz)exp$^{(i(k_{x}x+k_{y}y+k_{z}z-wt)}$ ,
where k = (kx,ky,kz),
to show that kXE=wB.

So I'm mainly fine with the method. I can see the maxwell's equaion ∇XE=-dB/dt, is the equation required.
-dB/dt=iwB.
And using ∇XE=ikXE, [1], the result follows.
My question is identifying equation [1]. How do you deduce this? Is it supposed to be obvious in any way. (I've done a check on the LHS and RHS so I can see its true), but should this be obvious?

Thanks in advance for your assistance .

 

[BvU]

Deduce, deduce? How would you write out ∇xE if ∇= $( {d\over dx}, {d\over dy}, {d\over dz})$ and $\vec E = \Bigl( E_x \exp ({i(k_{x}x+k_{y}y+k_{z}z-wt)}) , E_y \exp(i(k_{x}x+k_{y}y+k_{z}z-wt)) , E_y \exp(i(k_{x}x+k_{y}y+k_{z}z-wt))\Bigr)$ where $\vec E =(E_x, E_y, E_z)$ is a constant vector.
You don't even have to carry out the cross product. Only the diferentiation.

 

[binbagsss]

I'm not seeing it to be honest. I'm writing out ∇xE in determinant form.

 

[BvU]

Determinant form has $$\left \vert \matrix{\hat \imath&\hat \jmath&\hat k\cr {d\over dx}&{d\over dy}&{d\over dz}\cr E_x &E_y &E_z }\right \vert $$ if I remember well.
Write out e.g the x component and compare with the x-component of $\vec k\times\vec E$, etc.

 

[binbagsss]

In the bottom line of my first post - I've done this. But it's still not obvious to me until I do this?

 

[BvU]

Ah, I see. You did it, you see it's correct by inspection, but apparently you need something more to be really convinced ?

I can see the maxwell's equation ∇XE=-dB/dt, is the equation required.

Looks to me as if you are interpreting "required" somewhat unusual.

My interpretation of the exercise is more like: here we have an expression for $\vec E(\vec x,t)$ and $\vec B(\vec x,t)$ that we propose as a solution for Faraday's law (i.e. one of the Maxwell equations). And you are asked to make a few steps towards confirming it satisfies this equation -- under certain conditions for (Ex,Ey,Ez), (Bx,By,Bz), $\vec k$ and $\omega$.