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Chapter 2: Problem 134

The area of the triangle in the Arg and diagram formed by the Complex number \(\mathrm{z}\), iz and \(\mathrm{z}+\mathrm{iz}\) is (a) \(|z|^{2}\) (b) \((\sqrt{3} / 2)|z|^{2}\) (c) \((1 / 2)|\mathrm{z}|^{2}\) (d) \((3 / 2)|\mathrm{z}|^{2}\)

Short Answer

Expert verified
The area of the triangle is \(\frac{1}{2}|z|^2\), which corresponds to option (c).

Step by step solution

01

Convert complex numbers to vectors

First, we need to convert the complex numbers into vectors in the complex plane. Let's denote the complex number z as \(z = x + yi\), where x and y are real numbers and i is the imaginary unit. Then, the three vertices of the triangle are: 1. z = (x, y) 2. iz = (-y, x) 3. z + iz = (x-y, x+y)
02

Calculate vector for each side of the triangle

Now, we need to find the vectors representing the sides of the triangle. These vectors can be found by subtracting the coordinates of one vertex from another: Vector A = iz - z = (-y-x, x-y) Vector B = (z+iz) - iz = (x, y)
03

Determinant formula for area

The area of the triangle can be calculated using the following formula for the determinant of these vectors: Area = \(\frac{1}{2}|\text{det}|(A, B)|\) The determinant of matrix A and B is the following: \( \text{det} = \text{ABS} [ (x)(x-y) - (y)(-y-x) ] \)
04

Calculate area in terms of z

Now, we can plug in the values of x and y to calculate the area in terms of z: \( \text{Area} = \frac{1}{2}|\text{det}| = \frac{1}{2} | x^{2} - xy + xy + y^{2} | = \frac{1}{2} | x^2 + y^2 | \) Since the magnitude of the complex number z is given by \( |z|^2 = x^2 + y^2 \), we get: Area = \(\frac{1}{2} |z|^2 \) This corresponds to option (c) in the given choices.

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Most popular questions from this chapter

If \(\mathrm{f}(\mathrm{x})=4 \mathrm{x}^{5}+5 \mathrm{x}^{4}-8 \mathrm{x}^{3}+5 \mathrm{x}^{2} 4 \mathrm{x}-34 \mathrm{i}\) and \(\mathrm{f}[(-1+\sqrt{3} \mathrm{i}) / 2]=\mathrm{a}+\mathrm{ib}\) then \(\mathrm{a}: \mathrm{b}=\) (a) \(1: 2\) (b) \(-2: 1\) (c) \(17: 1\) (d) \(-17: 1\)

If \(|z-4 / z|=2\) then the maximum value of \(|z|\) is (a) \(\sqrt{5}+1\) (b) 2 (c) \(2+\sqrt{2}\) (d) \(\overline{\sqrt{3}+1}\)

lt \(z\) be a complex number and \(|z+3|<8 .\) Then the value of \(|z-2|\) lies in (a) \([-2,13]\) (b) \([0,13]\) (c) \([2,13]\) (d) \([-13,2]\)

It \(\mathrm{z}^{2}+\mathrm{z}+1=0\) where \(\mathrm{z}\) is a complex number, then the value of \([z+(1 / z)]^{2}+\left[z^{2}+\left(1 / z^{2}\right)\right]^{2}+\left[z^{3}+\left(1 / z^{3}\right)\right]^{2}+\ldots\) \(+\left[z^{6}+\left(1 / z^{6}\right)\right]^{2}\) is (a) 18 (b) 54 (c) 6 (d) 12

For any integer \(\mathrm{n}, \arg \left[(\sqrt{3}+\mathrm{i})^{4 n+1} /(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}\right]=\) (a) \((\pi / 3)\) (b) \((\pi / 6)\) (c) \((2 \pi / 3)\) (d) \((5 \pi / 6)\)
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