which statement describes how to geometrically divide a complex number, z, by a second complex number…

which statement describes how to geometrically divide a complex number, z, by a second complex number, w?\nscale z by the modulus of w, then rotate clockwise by the argument of w.\nscale z by the modulus of w, then rotate counterclockwise by the argument of w.\nscale z by the reciprocal of the modulus of w, then rotate clockwise by the argument of w.\nscale z by the reciprocal of the modulus of w, then rotate counterclockwise by the argument of w.

which statement describes how to geometrically divide a complex number, z, by a second complex number, w?\nscale z by the modulus of w, then rotate clockwise by the argument of w.\nscale z by the modulus of w, then rotate counterclockwise by the argument of w.\nscale z by the reciprocal of the modulus of w, then rotate clockwise by the argument of w.\nscale z by the reciprocal of the modulus of w, then rotate counterclockwise by the argument of w.

Answer

Explanation:

Step1: Recall division formula for complex numbers in polar form

If (z = r_1(\cos\theta_1 + i\sin\theta_1)) and (w=r_2(\cos\theta_2 + i\sin\theta_2)), then (\frac{z}{w}=\frac{r_1}{r_2}[\cos(\theta_1-\theta_2)+i\sin(\theta_1 - \theta_2)])

Step2: Analyze the modulus and argument operations

The modulus of (\frac{z}{w}) is (\frac{r_1}{r_2}) (scaling (z) by the reciprocal of the modulus of (w), since (\frac{r_1}{r_2}=r_1\times\frac{1}{r_2})). The argument of (\frac{z}{w}) is (\theta_1-\theta_2), which is equivalent to rotating (z) (whose argument is (\theta_1)) clockwise by the argument (\theta_2) of (w) (because subtracting an angle in the complex - plane rotation is equivalent to clockwise rotation when considering the formula for division in polar form).

Answer:

Scale (z) by the reciprocal of the modulus of (w), then rotate clockwise by the argument of (w).