I come across A Visual, Intuitive Guide to Imaginary Numbers, and find it's difficult for me to understand this section:
Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What’s the new heading?
……
Let’s try a simpler approach: we’re on a heading of 3 + 4i (whatever that angle is; we don’t really care), and want to rotate by 45 degrees. Well, 45 degrees is 1 + i (perfect diagonal), so we can multiply by that amount!
Per my understanding, rotation should not change the distance from 0, but it's obviously the distance of 3 + 4i isn't same with -1 + 7j. How to understand complex rotation?
Best Answer
They are only interested in the angle. If you want to preserve the magnitude, you must multiply by a complex number of absolute value one; in this example, you should take $(1+i)/\sqrt2$.