Prove the result of multiplying a complex number by (1 + i)

Question

I know that if I multiply a complex number that has the form a + bi by the number (1 + i), this will rotate the vector that corresponds to the complex number by 45 degrees in the counterclockwise direction, and also it will change the length of the original vector by scale factor $\sqrt2$. That is, given a vector z = a + bi which has length |z|, my new vector will have length $\sqrt2$|z|. I also recognize that if I graph the vector that represents 1 + i, I have a vector that has a 45 degree angle with respect to the real axis, and it has length of $\sqrt2$ so it makes sense that multiplying this vector by z=a+bi will give me a vector that is rotated 45 degrees and has length $\sqrt2$ |z|. However, I need to prove this geometrically, not algebraically. Can someone give a suggestion of how to do this?

Answer 1

Notice that $$i(a+bi)=-b+ai$$ which is perpendicular to (and has the same length as) $a+bi$ .

Then $$(a+bi)(1+i)=(a+bi)+i(a+bi)$$

which means that we are adding to $a+bi$ a vector of the same length as $a+bi$ in a perpendicular direction. The geometry regarding the length and angle follows neatly from there.