Math Problem Statement

Čemu je jednako |w| ako je w = \(\frac{\bar{z}}{z^2}\) i z = 3i - 3?

Solution

The problem in the image asks:

"Čemu je jednako |w| ako je w=zˉz2w = \frac{\bar{z}}{z^2} i z=3i3z = 3i - 3?"

This translates to:

"What is the modulus w|w| if w=zˉz2w = \frac{\bar{z}}{z^2} and z=3i3z = 3i - 3?"

Let's solve this step-by-step.

  1. Determine zz:
    We are given z=3i3z = 3i - 3. This is a complex number, so we write it as z=3+3iz = -3 + 3i.

  2. Find the conjugate of zz:
    The conjugate of zz, denoted as zˉ\bar{z}, is obtained by changing the sign of the imaginary part. Thus,
    zˉ=33i\bar{z} = -3 - 3i

  3. Calculate z2z^2:
    Now, calculate z2z^2: z2=(3+3i)2z^2 = (-3 + 3i)^2 Using the formula (a+bi)2=a2+2abi+(bi)2(a + bi)^2 = a^2 + 2abi + (bi)^2, we get: (3+3i)2=(3)2+2(3)(3i)+(3i)2=918i+9i2(-3 + 3i)^2 = (-3)^2 + 2(-3)(3i) + (3i)^2 = 9 - 18i + 9i^2 Since i2=1i^2 = -1, this simplifies to: z2=918i9=18iz^2 = 9 - 18i - 9 = -18i

  4. Substitute into ww:
    Now, substitute zˉ\bar{z} and z2z^2 into the expression for ww: w=zˉz2=33i18iw = \frac{\bar{z}}{z^2} = \frac{-3 - 3i}{-18i} Simplifying the denominator: w=33i18i=33i18iii=(33i)(i)18i2w = \frac{-3 - 3i}{-18i} = \frac{-3 - 3i}{-18i} \cdot \frac{-i}{-i} = \frac{(-3 - 3i)(-i)}{-18i^2} Since i2=1i^2 = -1, the denominator becomes: 18i2=18-18i^2 = 18 Now simplify the numerator: (33i)(i)=3i+3=3+3i(-3 - 3i)(-i) = 3i + 3 = 3 + 3i Thus, we have: w=3+3i18=16+16iw = \frac{3 + 3i}{18} = \frac{1}{6} + \frac{1}{6}i

  5. Find w|w|:
    The modulus of w=16+16iw = \frac{1}{6} + \frac{1}{6}i is given by: w=(16)2+(16)2=136+136=236=118=118=132=26|w| = \sqrt{\left( \frac{1}{6} \right)^2 + \left( \frac{1}{6} \right)^2} = \sqrt{\frac{1}{36} + \frac{1}{36}} = \sqrt{\frac{2}{36}} = \sqrt{\frac{1}{18}} = \frac{1}{\sqrt{18}} = \frac{1}{3\sqrt{2}} = \frac{\sqrt{2}}{6}

Thus, the modulus of ww, w|w|, is 26\frac{\sqrt{2}}{6}.


Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions you might find interesting:

  1. How do you find the conjugate of a complex number?
  2. What is the geometric interpretation of the modulus of a complex number?
  3. How can you simplify complex fractions involving imaginary units?
  4. What are the properties of the modulus function in complex numbers?
  5. How do you square a complex number using the standard form a+bia + bi?

Tip: Always remember that multiplying the numerator and denominator of a fraction by the conjugate is a useful method to simplify expressions with complex numbers.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Conjugates
Modulus of Complex Numbers
Algebraic Manipulation

Formulas

Conjugate of z: \(\bar{z}\)
Modulus formula: \(|w| = \sqrt{(\text{Re}(w))^2 + (\text{Im}(w))^2}\)
Square of complex numbers: \((a + bi)^2 = a^2 + 2abi + b^2i^2\)

Theorems

Modulus of a Complex Number
Conjugate Property in Complex Numbers

Suitable Grade Level

Grades 11-12 (Pre-Calculus, Algebra II)