Question

Hi, can you please help me solve this question part (e)?

Answer 1

Given the vector

$z=-3+4i$

Part A

The modulus can be found as follow

$\sqrt[]{(-3)^2+4^2}=\sqrt[]{9+16}=\sqrt[]{25}=5$

$|z|=5$

Part B.

The argument of z can be found as follow

$\begin{gathered} \tan \text{ }\emptyset=\frac{4}{3} \\ \emptyset=\tan ^{-1}(\frac{4}{3}) \\ \emptyset=53.130^0 \end{gathered}$

Since it is in the second quadrant,

$\emptyset=180^0-53.130^0=126.870^0$

Part C

$w=\frac{10+4i}{z}$

$w=\frac{10+4i}{-3+4i}$

we will rationalize the denominator

$\begin{gathered} w=\frac{10+4i}{-3+4i}\times\frac{-3-4i}{-3-4i} \\ \\ w=\frac{10+4i}{9+16}=\frac{-30-12i-40i+16}{25}=\frac{-14-52i}{25} \\ \\ w=\frac{-14}{25}-\frac{52i}{25} \\ \end{gathered}$