How do arrows add up, and why is that interference?
To add two arrows, put the tail of the second one on the tip of the first. The sum is the arrow from the start of the first to the tip of the second. In parts, you just add the two parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.
Arrows pointing the same way add their lengths. Arrows pointing in opposite directions subtract. At any other angle, the result is somewhere in between.
Two arrows of length 1 point in exactly opposite directions. How long is their sum? The second arrow walks back exactly as far as the first went: you end where you started, and the sum has length 0.
Now think of two waves arriving at the same place, each drawn as an arrow. The angle between the arrows is the phase difference between the waves. Adding the arrows gives the combined wave: its length is the combined amplitude, and its square is the brightness.
In phase, two arrows of length 1 add to length 2: brightness 4. Opposite, they add to 0: darkness. This is exactly the interference of module 2, now as simple geometry. Feynman summed up the rule of quantum physics in three steps: for each way something can happen, draw an arrow; add the arrows; square the length of the result. The last lesson of this module puts the rule to work.
|z + w|^2 = |z|^2 + |w|^2 + 2 |z| |w| cos φ, where φ is the angle between the arrows.
Two arrows of length 1 at angle φ: |sum| = 2 |cos(φ/2)|, and |sum|^2 = 4 cos^2(φ/2) = 2 + 2 cos φ. The same curve as the fringes of module 2.
The middle term, 2 |z| |w| cos φ, is the interference term: positive when the arrows roughly agree, negative when they oppose.
Adding arrows tip to tail is interference in geometry: aligned arrows reinforce, opposite arrows cancel, and the square of the length is the brightness.