Alright, let’s solve for the magnitude of the complex number [tex]\( z = \frac{1 - i \sqrt{3}}{2 - 2i} \)[/tex].
First, we recognize that the magnitude of a complex number [tex]\( z = a + bi \)[/tex] is given by [tex]\( |z| = \sqrt{a^2 + b^2} \)[/tex].
Given the complex number [tex]\( z \)[/tex], we need to simplify it first. To do this, we'll multiply the numerator and the denominator of [tex]\( z \)[/tex] by the conjugate of the denominator.
The conjugate of [tex]\( 2 - 2i \)[/tex] is [tex]\( 2 + 2i \)[/tex]. Therefore, we have:
[tex]\[ z = \frac{1 - i \sqrt{3}}{2 - 2i} \cdot \frac{2 + 2i}{2 + 2i} \][/tex]
Now, let's multiply the numerators and the denominators:
[tex]\[ \frac{(1 - i \sqrt{3})(2 + 2i)}{(2 - 2i)(2 + 2i)} \][/tex]
First, simplify the denominator:
[tex]\[ (2 - 2i)(2 + 2i) = 4 + 4i - 4i - 4i^2 \][/tex]
[tex]\[ = 4 + 4 \][/tex]
[tex]\[ = 8 \][/tex]
Now, simplify the numerator:
[tex]\[ (1 - i \sqrt{3})(2 + 2i) = 2 + 2i - 2i \sqrt{3} - 2i^2 \sqrt{3} \][/tex]
[tex]\[ = 2 + 2i - 2i \sqrt{3} + 2 \sqrt{3} \quad \text{(since } i^2 = -1 \text{)} \][/tex]
So, we have:
[tex]\[ \frac{2 + 2i - 2i \sqrt{3} + 2 \sqrt{3}}{8} \][/tex]
Simplifying the numerator further:
[tex]\[ 2\frac{1 + \sqrt{3}}{8} + 2i \frac{(1 - \sqrt{3})}{8} \][/tex]
[tex]\[ = \frac{1 + \sqrt{3}}{4} + i \frac{(1 - \sqrt{3})}{4} \][/tex]
Let's denote the simplified form of [tex]\( z \)[/tex] as [tex]\( \frac{a}{4} + i \frac{b}{4} \)[/tex], where [tex]\( a = 1 + \sqrt{3} \)[/tex] and [tex]\( b = 1 - \sqrt{3} \)[/tex].
Now, solving for the magnitude:
[tex]\[ |z| = \sqrt{\left(\frac{1 + \sqrt{3}}{4}\right)^2 + \left(\frac{1 - \sqrt{3}}{4}\right)^2} \][/tex]
[tex]\[ = \frac{1}{4}\sqrt{(1 + \sqrt{3})^2 + (1 - \sqrt{3})^2} \][/tex]
[tex]\[ = \frac{1}{4}\sqrt{(1 + 2\sqrt{3} + 3) + (1 - 2\sqrt{3} + 3)} \][/tex]
[tex]\[ = \frac{1}{4}\sqrt{(4 + 2 \sqrt{3}) + (4 - 2 \sqrt{3})} \][/tex]
[tex]\[ = \frac{1}{4}\sqrt{8} \][/tex]
[tex]\[ = \frac{1}{4} \times \sqrt{8} \][/tex]
[tex]\[ = \frac{\sqrt{8}}{4} \][/tex]
[tex]\[ = \frac{2 \sqrt{2}}{4} \][/tex]
[tex]\[ = \frac{\sqrt{2}}{2} \][/tex]
Therefore:
[tex]\[ |z| = \boxed{0.7071067811865475} \][/tex]
