"Adding the arrow" part was confusing for me.

The sum of 2 arrows is done by placing the base of the second arrow on the tip of the first arrow. Then you draw a third arrow from the base of the first arrow towards the tip of the second arrow. That third arrow you drew is the sum of those 2 arrows.

The red arrow in this example would be the final arrow.

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how/why would a wave cause both to go off?

The "wave front" would hit all areas at once. Think of a wave on a beach, how many places at once it hits? Almost the whole beach at the same time.

Hopefully this was clear and didn't get too much distracted laughing at my illustrations. 😅