Phasors: waves as spinning arrows

Two spinning arrows add tip to tail, and their shadows add up to one wave.

Adding waves with spinning arrows

Interactive

amplitudes1 + 1

phase difference90°

sum's amplitude1.41

Second wave's amplitude1Phase difference φ90 °

The math, with your numbers

\cos\omega t + 1\cos(\omega t + 90^\circ) = \mathrm{Re}\big[(1 + 1\,e^{i\,90^\circ})\,e^{i\omega t}\big]

A = |1 + 1\,e^{i\varphi}| = \sqrt{1 + 1 + 2(1)\cos 90^\circ} = 1.41

Try this

→Equal amplitudes: which phase difference doubles the wave, and which wipes it out?
→At 90°, two waves of amplitude 1 add to what amplitude? Look at the triangle of arrows.
→Pause and find the moment when the rose wave is at its peak. Where is the rose arrow pointing?

Want to know why it behaves like this?

The lesson “Euler's Formula & Phasors” explains it step by step: Why do e, i and π belong in one equation?