Chapter 9.6, Problem 50E

To determine

To find the product of $z_1{z}_2$

Answer to Problem 50E

  $\left(\sqrt{3}-1\right)\left(1+\sqrt{3}\right)i$

Explanation of Solution

Given:

The sample example taken is $z_1=\sqrt{3}+i$ and $z_2=1+i$

Calculation:

The product

  $\begin{array}{l}{z}_1{z}_2=\left(\sqrt{3}+i\right)\left(1+i\right)\\ =\sqrt{3}+\sqrt{3}i+i-i^2\\ =\left(\sqrt{3}-1\right)\left(1+\sqrt{3}\right)i\end{array}$

To determine

To write $z_1,z_2$ and $z_1{z}_2$ in polar form.

Answer to Problem 50E

  $z_1{z}_2=2\sqrt{2}(\cos {75}^{\circ }+i\sin {75}^{\circ })$

Explanation of Solution

Given:

The sample example taken is $z_1=\sqrt{3}+i$ and $z_2=1+i$

Calculation:

The polar form of $z_1=\frac{1}{2}(\cos {30}^{\circ }+i\sin {30}^{\circ })$

The polar form of $z_2=\sqrt{2}(\cos {45}^{\circ }+i\sin {45}^{\circ })$

The polar form of $z_1{z}_2$

  $\begin{array}{l}{z}_1{z}_2=\frac{1}{2}(\cos {30}^{\circ }+i\sin {30}^{\circ })\sqrt{2}(\cos {45}^{\circ }+i\sin {45}^{\circ })\\ z_1{z}_2=2\sqrt{2}(\cos {75}^{\circ }+i\sin {75}^{\circ })\end{array}$

To determine

To repeat the procedure with different pair of complex numbers.

Answer to Problem 50E

Dilation by a factor of $3$ .

Explanation of Solution

Given:

The given operation is $w=3z$

Calculation:

The transformation $w=3z$ dilates $z$ by a factor of $3$ .

To determine

To make a conjecture about the product of two complex numbers in polar form.

Explanation of Solution

From previous part we can conclude that to multiply two complex numbers in polar form,

multiply the modulus and add the amplitudes.