Aus Online Mathematik Brückenkurs 2

If we calculate the expression, we get the answer at once,

\displaystyle \begin{align} z-\bar{w}+u &= (2+i)-(2-3i)+(-1-2i)\\[5pt] &= 2-2-1+(1+3-2)i\\[5pt] &= -1+2i\,\textrm{.} \end{align}

If, on the other hand, we interpret the expression in terms of vectors, we must first understand the vector \displaystyle \bar{w} geometrically. When we take the complex conjugate of \displaystyle w, we change the sign of the imaginary part, which is the same as reflecting \displaystyle w in the real axis.

We can then construct the expression \displaystyle z-\bar{w}+u one term at a time.