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Sagot :
Sure, let's solve these problems step-by-step.
### (a) Verify [tex]\(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)[/tex]
Given:
[tex]\[ z_1 = 6 + 3i \][/tex]
[tex]\[ z_2 = 5 - 2i \][/tex]
1. Calculate the difference [tex]\( z_1 - z_2 \)[/tex]:
[tex]\[ z_1 - z_2 = (6 + 3i) - (5 - 2i) \][/tex]
[tex]\[ z_1 - z_2 = 6 + 3i - 5 + 2i \][/tex]
[tex]\[ z_1 - z_2 = 1 + 5i \][/tex]
2. Find the conjugate of the difference [tex]\( z_1 - z_2 \)[/tex]:
[tex]\[ \overline{z_1 - z_2} = \overline{1 + 5i} \][/tex]
[tex]\[ \overline{z_1 - z_2} = 1 - 5i \][/tex]
3. Find the conjugates of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[ \overline{z_1} = \overline{6 + 3i} = 6 - 3i \][/tex]
[tex]\[ \overline{z_2} = \overline{5 - 2i} = 5 + 2i \][/tex]
4. Calculate the difference of the conjugates:
[tex]\[ \overline{z_1} - \overline{z_2} = (6 - 3i) - (5 + 2i) \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 6 - 3i - 5 - 2i \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 1 - 5i \][/tex]
5. Verify that [tex]\(\overline{z_1 - z_2}\)[/tex] equals [tex]\(\overline{z_1} - \overline{z_2}\)[/tex]:
[tex]\[ \overline{z_1 - z_2} = 1 - 5i \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 1 - 5i \][/tex]
Since both are equal, the verification shows that [tex]\(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)[/tex].
### (b) Find [tex]\(\arg(z_1)\)[/tex]
Given:
[tex]\[ z_1 = 6 + 3i \][/tex]
1. Find the argument of [tex]\( z_1 \)[/tex]:
The argument [tex]\( \arg(z_1) \)[/tex] is the angle [tex]\( \theta \)[/tex] in the complex plane such that:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) \][/tex]
[tex]\[ \theta = \tan^{-1}\left(\frac{3}{6}\right) \][/tex]
[tex]\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \][/tex]
We find:
[tex]\[ \theta \approx 0.4636476090008061 \][/tex]
Thus, [tex]\(\arg(z_1) \approx 0.4636476090008061\)[/tex].
These steps show the verification of the given properties of the complex numbers and the calculation of the argument for [tex]\( z_1 \)[/tex].
### (a) Verify [tex]\(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)[/tex]
Given:
[tex]\[ z_1 = 6 + 3i \][/tex]
[tex]\[ z_2 = 5 - 2i \][/tex]
1. Calculate the difference [tex]\( z_1 - z_2 \)[/tex]:
[tex]\[ z_1 - z_2 = (6 + 3i) - (5 - 2i) \][/tex]
[tex]\[ z_1 - z_2 = 6 + 3i - 5 + 2i \][/tex]
[tex]\[ z_1 - z_2 = 1 + 5i \][/tex]
2. Find the conjugate of the difference [tex]\( z_1 - z_2 \)[/tex]:
[tex]\[ \overline{z_1 - z_2} = \overline{1 + 5i} \][/tex]
[tex]\[ \overline{z_1 - z_2} = 1 - 5i \][/tex]
3. Find the conjugates of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[ \overline{z_1} = \overline{6 + 3i} = 6 - 3i \][/tex]
[tex]\[ \overline{z_2} = \overline{5 - 2i} = 5 + 2i \][/tex]
4. Calculate the difference of the conjugates:
[tex]\[ \overline{z_1} - \overline{z_2} = (6 - 3i) - (5 + 2i) \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 6 - 3i - 5 - 2i \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 1 - 5i \][/tex]
5. Verify that [tex]\(\overline{z_1 - z_2}\)[/tex] equals [tex]\(\overline{z_1} - \overline{z_2}\)[/tex]:
[tex]\[ \overline{z_1 - z_2} = 1 - 5i \][/tex]
[tex]\[ \overline{z_1} - \overline{z_2} = 1 - 5i \][/tex]
Since both are equal, the verification shows that [tex]\(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\)[/tex].
### (b) Find [tex]\(\arg(z_1)\)[/tex]
Given:
[tex]\[ z_1 = 6 + 3i \][/tex]
1. Find the argument of [tex]\( z_1 \)[/tex]:
The argument [tex]\( \arg(z_1) \)[/tex] is the angle [tex]\( \theta \)[/tex] in the complex plane such that:
[tex]\[ \theta = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) \][/tex]
[tex]\[ \theta = \tan^{-1}\left(\frac{3}{6}\right) \][/tex]
[tex]\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \][/tex]
We find:
[tex]\[ \theta \approx 0.4636476090008061 \][/tex]
Thus, [tex]\(\arg(z_1) \approx 0.4636476090008061\)[/tex].
These steps show the verification of the given properties of the complex numbers and the calculation of the argument for [tex]\( z_1 \)[/tex].
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