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Amazing set of notes! Highly recommend purchasing them :)

The shortest distance is the perpendicular distance, horizontal distance is 2 (From origin to z+2) and arg(z+2) = pi/4. You can also use trig ratios and angle sum of triangle to solve for the |z| and arg(z). Then put it into mod-arg form. Rcis(theta). Convert to x+iy form.

Amazing set of physics notes. Worth the price.

I just bought the Chem notes. They are absolutely worth every penny. Thank you Kamal :)

You're missing the "y ≠ 0" condition. If Re(w) = 0, y = 0, then it is Real not Imaginary. 0 + 0i = Real.