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Sagot :
To solve the problem of finding the products and quotients of the given complex numbers [tex]\( z \)[/tex] and [tex]\( w \)[/tex], we first need to understand how to work with complex numbers and their representations in different forms.
The numbers given are:
[tex]\[ z = 11 - 11i \][/tex]
[tex]\[ w = \sqrt{3} + i \][/tex]
### Step 1: Multiply [tex]\( z \)[/tex] and [tex]\( w \)[/tex]
First, compute the product [tex]\( z \times w \)[/tex]:
[tex]\[ z \times w = (11 - 11i) \times (\sqrt{3} + i) \][/tex]
### Step 2: Convert the product to polar form
The result of the multiplication [tex]\( z \times w \)[/tex] when converted to polar form gives us a magnitude and an angle:
[tex]\[ |z \times w| = 31.11269837220809 \][/tex]
[tex]\[ \arg(z \times w) = -0.2617993877991494 \][/tex]
So, the polar form of [tex]\( z \times w \)[/tex] is:
[tex]\[ z \times w \approx 31.11269837220809 \angle -0.2617993877991494 \][/tex]
### Step 3: Write the product in exponential form
We convert the polar form to exponential form using Euler's formula:
[tex]\[ z \times w = 31.11269837220809 \cdot e^{-0.2617993877991494i} \][/tex]
### Step 4: Divide [tex]\( z \)[/tex] by [tex]\( w \)[/tex]
Now, compute the quotient [tex]\( \frac{z}{w} \)[/tex]:
[tex]\[ \frac{z}{w} = \frac{11 - 11i}{\sqrt{3} + i} \][/tex]
### Step 5: Convert the quotient to polar form
The result of the division [tex]\( \frac{z}{w} \)[/tex] when converted to polar form gives us a magnitude and an angle:
[tex]\[ \left|\frac{z}{w}\right| = 7.778174593052023 \][/tex]
[tex]\[ \arg \left( \frac{z}{w} \right) = -1.3089969389957472 \][/tex]
So, the polar form of [tex]\( \frac{z}{w} \)[/tex] is:
[tex]\[ \frac{z}{w} \approx 7.778174593052023 \angle -1.3089969389957472 \][/tex]
### Step 6: Write the quotient in exponential form
We convert the polar form to exponential form using Euler's formula:
[tex]\[ \frac{z}{w} = 7.778174593052023 \cdot e^{-1.3089969389957472i} \][/tex]
### Summary
1. The product [tex]\( z \times w \)[/tex] in polar form is:
[tex]\[ 31.11269837220809 \angle -0.2617993877991494 \][/tex]
In exponential form:
[tex]\[ 31.11269837220809 \cdot e^{-0.2617993877991494i} \][/tex]
2. The quotient [tex]\( \frac{z}{w} \)[/tex] in polar form is:
[tex]\[ 7.778174593052023 \angle -1.3089969389957472 \][/tex]
In exponential form:
[tex]\[ 7.778174593052023 \cdot e^{-1.3089969389957472i} \][/tex]
The numbers given are:
[tex]\[ z = 11 - 11i \][/tex]
[tex]\[ w = \sqrt{3} + i \][/tex]
### Step 1: Multiply [tex]\( z \)[/tex] and [tex]\( w \)[/tex]
First, compute the product [tex]\( z \times w \)[/tex]:
[tex]\[ z \times w = (11 - 11i) \times (\sqrt{3} + i) \][/tex]
### Step 2: Convert the product to polar form
The result of the multiplication [tex]\( z \times w \)[/tex] when converted to polar form gives us a magnitude and an angle:
[tex]\[ |z \times w| = 31.11269837220809 \][/tex]
[tex]\[ \arg(z \times w) = -0.2617993877991494 \][/tex]
So, the polar form of [tex]\( z \times w \)[/tex] is:
[tex]\[ z \times w \approx 31.11269837220809 \angle -0.2617993877991494 \][/tex]
### Step 3: Write the product in exponential form
We convert the polar form to exponential form using Euler's formula:
[tex]\[ z \times w = 31.11269837220809 \cdot e^{-0.2617993877991494i} \][/tex]
### Step 4: Divide [tex]\( z \)[/tex] by [tex]\( w \)[/tex]
Now, compute the quotient [tex]\( \frac{z}{w} \)[/tex]:
[tex]\[ \frac{z}{w} = \frac{11 - 11i}{\sqrt{3} + i} \][/tex]
### Step 5: Convert the quotient to polar form
The result of the division [tex]\( \frac{z}{w} \)[/tex] when converted to polar form gives us a magnitude and an angle:
[tex]\[ \left|\frac{z}{w}\right| = 7.778174593052023 \][/tex]
[tex]\[ \arg \left( \frac{z}{w} \right) = -1.3089969389957472 \][/tex]
So, the polar form of [tex]\( \frac{z}{w} \)[/tex] is:
[tex]\[ \frac{z}{w} \approx 7.778174593052023 \angle -1.3089969389957472 \][/tex]
### Step 6: Write the quotient in exponential form
We convert the polar form to exponential form using Euler's formula:
[tex]\[ \frac{z}{w} = 7.778174593052023 \cdot e^{-1.3089969389957472i} \][/tex]
### Summary
1. The product [tex]\( z \times w \)[/tex] in polar form is:
[tex]\[ 31.11269837220809 \angle -0.2617993877991494 \][/tex]
In exponential form:
[tex]\[ 31.11269837220809 \cdot e^{-0.2617993877991494i} \][/tex]
2. The quotient [tex]\( \frac{z}{w} \)[/tex] in polar form is:
[tex]\[ 7.778174593052023 \angle -1.3089969389957472 \][/tex]
In exponential form:
[tex]\[ 7.778174593052023 \cdot e^{-1.3089969389957472i} \][/tex]
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