Let $O$ be the origin, the point $A$ be $z_1 = \sqrt{3} + 2 \sqrt{2} i$, the point $B(z_2)$ be such that $\sqrt{3} |z_2| = |z_1|$ and $\arg(z_2) = \arg(z_1) + \frac{\pi}{2}$. Then

Step 1: Calculate Modulus and Argument of $z_1$

First, we need to find the modulus and argument of the complex number $z_1$. The modulus of a complex number $a+bi$ is given by $\sqrt{a^2+b^2}$, and its argument is $\tan^{-1}(b/a)$. For $z_1 = \sqrt{3} + 2\sqrt{2}i$, we calculate its modulus as $\sqrt{(\sqrt{3})^2 + (2\sqrt{2})^2} = \sqrt{3+8} = \sqrt{11}$. The argument is $\tan^{-1}\left(\frac{2\sqrt{2}}{\sqrt{3}}\right)$.

Step 2: Calculate Modulus of $z_2$

We are given the relation $\sqrt{3} |z_2| = |z_1|$. We can use the modulus of $z_1$ we just calculated to find the modulus of $z_2$. Dividing both sides by $\sqrt{3}$, we get $|z_2| = \frac{\sqrt{11}}{\sqrt{3}}$, which simplifies to $\sqrt{\frac{11}{3}}$.

Step 3: Calculate Argument of $z_2$

We are given that the argument of $z_2$ is $\frac{\pi}{2}$ radians (or 90 degrees) greater than the argument of $z_1$. This means that the vector representing $z_2$ is obtained by rotating the vector representing $z_1$ counter-clockwise by 90 degrees.

Step 4: Express $z_2$ in terms of $z_1$

We can express $z_2$ in polar form using its modulus and argument. Since $\arg(z_2) = \arg(z_1) + \frac{\pi}{2}$, we can write $e^{i \arg(z_2)} = e^{i \arg(z_1)} e^{i \frac{\pi}{2}}$. Also, we know that $e^{i \frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) = 0 + i(1) = i$. Substituting these into the polar form for $z_2$, we get $z_2 = \frac{|z_1|}{\sqrt{3}} e^{i \arg(z_1)} i$. Since $z_1 = |z_1| e^{i \arg(z_1)}$, we can write $z_2 = \frac{i}{\sqrt{3}} z_1$.

Step 5: Substitute $z_1$ to find $z_2$

Now we substitute the given value of $z_1 = \sqrt{3} + 2\sqrt{2}i$ into the expression for $z_2$. We multiply $\frac{i}{\sqrt{3}}$ by $z_1$. Remember that $i^2 = -1$. After distributing and simplifying, we get $z_2 = -\frac{2\sqrt{2}}{\sqrt{3}} + i$.

More from this chapter

• Express (1 + i)^4 using De Moivre's theorem.
• Simplify (1 + ²)^5.
• Simplify (1-)/(1- ²).
• **LONG ANSWER TYPE QUESTIONS** 1) If α and β are different complex numbers with |β| = 1, then find | β - α1 - αβ| 2) If |a + ib| = 1 then prove that ((1 + b + i
• Find all values of (1+i)^(1-i) in the form a + bi.
• Find all real solutions to the equation: √(x + 3 - 4√(x - 1)) + √(x + 8 - 6√(x - 1)) = 1.
• If the set of all a R, for which the equation 2x²+(a-5)x+15=3a has no real root, is the interval (α, β), and X=x Z: α < x < β, then Σ_x X x² is equal to:
• MODIFIED: Given that variable_91, variable_91, variable_91 are roots of x³ - 3x² + 4 = 0, find the value of variable_91⁴ + variable_91⁴ + variable_91⁴.

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