Problem (with solution) about Heisenberg's microscope

s3a

1. The problem statement, all variables and given/known data
The problem along with its solution are included as ProblemSolution.jpg.

2. Relevant equations
(Eq. 1) Δx ≈ λ/(sinθ)
(Eq. 2) Δp_x ≈ 2h(v/c)(sinθ) = 2(h/λ)(sinθ)
(Eq. 3) Δx Δp = ħ/2

3. The attempt at a solution
I am confused about several things. Firstly, geometrically, what is Δx and λ? I ask because I am trying to make mathematical sense of the Δx ≈ λ/(sinθ) equation rather than just memorizing it. I am not looking for any rigorous proof or anything unecessarily challenging but just an intuitive derivation.

Secondly, why does introducing the Δp_x ≈ 2h(v/c)(sinθ) = 2(h/λ)(sinθ) equation help in answering the question; that is, how does this equation “show that if we minimize Δx by reducing λ, this will result in a loss of information about the x-component of the electron momentum?”

Also, if the electrons and photons are in 100% horizontal (x-axis) motion, then why don't they simply slow down in the same direction or move in the opposite direction (still on the x-axis)? (The diagram implies that photons will move along the arrow-lines that point toward the lens.)

I also don't understand (1.5.3). Is (1.5.3) supposed be something like (Eq. 3) from above?

I would really like to understand this problem inside-out so any help in figuring this problem out will be greatly appreciated!

If more information is needed, just ask.

Thanks in advance!

Attached Thumbnails