/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 \(22 \mathrm{ABCD}\) is a square... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

\(22 \mathrm{ABCD}\) is a square, lettered anticlockwise, on an Argand diagram, with A representing \(3+\mathrm{j} 2\) and \(\mathrm{B}\) representing \(-1+j 4\). Show that \(\mathrm{C}\) lies on the real axis and find the complex number represented by D and the length of \(\mathrm{AB}\).

Short Answer

Expert verified
C is at \(-3+0j\), D is \(1-j2\); AB is \(2\sqrt{5}\).

Step by step solution

01

Understand the Properties of a Square in Argand Diagram

A square on an Argand diagram has sides of equal length and angles of 90 degrees. With the vertices labeled as A, B, C, and D anticlockwise, complex numbers at each vertex need to satisfy these properties. We are given the points A and B, and we know that points B to C, C to D, and D to A also represent equal distances.
02

Calculate Length of AB

The length between two complex numbers, for example from A to B, is the modulus (absolute distance) of the vector linking them. \[ AB = |(-1 + j4) - (3 + j2)| = |-4 + j2| = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
03

Establish the Relationship Between the Vertices

Since ABCD is a square, the sides are equal and the vectors AB, BC, CD, and DA are orthogonal. AB is already known, so to find C, the vector BC must be a 90-degree rotation of AB, which can be achieved by multiplying by \(j\). Thus, \( BC = j (B - A) \).
04

Determine Coordinates of Point C

Point C can be express as \( B + BC \):\[ B = -1 + j4, \, AB = -4 + j2 \]\[ BC = j(-4 + j2) = -2 - j4 \]\[ C = B + BC = (-1 + j4) + (-2 - j4) = -3 + j0 \]Thus, C lies on the real axis.
05

Determine Coordinates of Point D

The vector CD will also be a rotation of BC by 90 degrees, resulting in \(j(BC)\), hence similarly for DA. So to find D: \[ C = -3 + j0, \, CD = j(-2 - j4) = 4 - j2 \]\[ D = C + CD = (-3 + j0) + (4 - j2) = 1 - j2 \]
06

Verify and Conclude

The calculations confirm that C lies on the real axis at \(-3\). We found that D locates precisely at \(1 - j2\), ensuring our operations preserve equal vectors and right angles for the square. The length AB checked at \(2\sqrt{5}\), consistent for all sides, guarantees these solutions fulfill square's conditions.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Argand Diagram
An Argand diagram is a graphical representation of complex numbers. Each complex number corresponds to a unique point in the plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. This provides a powerful way to visualize complex numbers, which are typically written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

Key features of an Argand diagram include:
  • The real axis, which runs horizontally, showing the real parts of complex numbers.
  • The imaginary axis, running vertically, representing the imaginary parts.
  • Each point on the diagram can be seen as a vector originating from the origin \( (0,0) \) to the point \( (a,b) \).
Visualizing complex numbers on an Argand diagram allows us to easily perform operations such as addition, subtraction, and understanding geometric transformations like rotations and reflections.
Modulus of Complex Numbers
The modulus of a complex number is the equivalent of its "length" or "magnitude" in the context of the Argand diagram. For a complex number \( z = a + bi \), its modulus is given by the formula \( |z| = \sqrt{a^2 + b^2} \). This represents the distance of the point \( z \) from the origin \( (0,0) \) on the Argand diagram.

The modulus has several practical applications:
  • It helps determine the distance between two complex numbers by calculating the modulus of their difference.
  • It is used extensively in finding the magnitude of geometric transformations, as it remains unchanged under rotations.
  • The modulus is crucial in defining complex functions and their behavior.
In the problem of finding the length from A to B in a square, the distance AB was calculated using the modulus, ensuring that the length of each side is verified correctly.
Geometric Transformations
Geometric transformations involve changing the position or size of figures within a plane. In the context of complex numbers on an Argand diagram, these transformations can be achieved through operations that primarily involve rotation, translation, or reflection.

With complex numbers:
  • **Rotation:** You can rotate a complex number by multiplying it with another complex number of unit modulus (often involving \( i \) or a power of \( i \) to achieve 90-degree rotations).
  • **Translation:** This corresponds to adding a constant complex number to each point, effectively moving the entire shape.
  • **Reflection:** Achieved by conjugating the complex number, meaning flipping its sign in the imaginary part.
In the exercise, finding the positions of points C and D from A and B required rotating the complex vector representing side AB by 90 degrees using multiplication by \( i \). This ensures maintaining right-angle properties and equal length of sides, confirming the square's structure.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the line joining \((2,3,4)\) to \((1,2,3)\) is perpendicular to the line joining \((1,0,2)\) to \((2,3,-2)\)

A regular hexagon OACDEB has adjacent sides \(\overrightarrow{\mathrm{OA}}=a\) and \(\overrightarrow{\mathrm{OB}}=b\). Find the vectors \(\overrightarrow{\mathrm{OC}}, \overrightarrow{\mathrm{OD}}\) \(\overrightarrow{\mathrm{OE}}\) representing the other three corners in terms of \(a\) and \(b\).

(a) Determine the equation of the plane that passes through the points \((1,2,-2),(-1,1,-9)\) and \((2,-2,-12)\). Find the perpendicular distance from the origin to this plane. (b) Calculate the area of the triangle whose vertices are at the points \((1,1,0),(1,0,1)\) and \((0,1,1)\).

Two skew lines \(L_{1}, L_{2}\) have respective equations \(\frac{x+3}{4}=\frac{y-3}{-1}=\frac{z-2}{1}\) and \(\frac{x-1}{2}=\frac{y-5}{1}=\frac{z+3}{2}\) Obtain the equation of a plane through \(L_{1}\) parallel to \(L_{2}\) and show that the shortest distance between the lines is 6 .

Find the equation of the line joining \((1,-1,3)\) to \((3,3,-1)\). Show that it is perpendicular to the plane \(2 x+4 y-4 z=5\), and find the angle that the line makes with the plane \(12 x-15 y+16 z=10\)
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Rotational Dynamics

Read Explanation

Translational Dynamics

Read Explanation

Astrophysics

Read Explanation

Wave Optics

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.