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Gauß-Verfahren einfach erklärt: Beispiele, Übungen und Lösungen

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Gauß-Verfahren einfach erklärt: Beispiele, Übungen und Lösungen
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brn01

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The Gauß-Verfahren einfach erklärt and geometric relationships between lines and planes form the core of analytical geometry. This comprehensive guide covers solving linear equation systems, vector calculations, and analyzing positional relationships between geometric objects in three-dimensional space.

  • The Gauß-Verfahren Matrix is demonstrated through detailed examples showing step-by-step solutions
  • Vector operations include calculating Richtungsvektor und Stützvektor for lines and planes
  • Lagebeziehung Gerade Ebene concepts are explored through various scenarios including parallel, intersecting, and skew lines
  • Practical examples demonstrate how to determine if points lie on lines or planes
  • Special cases of Gauß-Verfahren unendlich viele Lösungen are explained with clear illustrations

19.10.2021

6030

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Vector Representation and Geometric Relationships

This page focuses on vector representation and geometric relationships in three-dimensional space. It introduces key concepts for understanding spatial relationships between points, lines, and planes.

Vocabulary:

  • Stützvektor (support vector): A vector that defines a point on a line or plane
  • Richtungsvektor (direction vector): A vector that indicates the direction of a line or plane

The page includes a diagram showing various points in 3D space, labeled from A to H, which helps visualize the spatial relationships discussed.

Highlight: Understanding the representation of vectors and points in 3D space is crucial for solving problems involving lines and planes.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Vector Operations and Triangle Properties

This page explores vector operations and their application in determining geometric properties of triangles. It presents a specific example of using vectors to check if a triangle is isosceles.

Example: Given points A(1,2,3), B(2,4,3), and C(3,1,3), the page demonstrates how to calculate vector lengths to determine if the triangle is isosceles.

The solution involves calculating the lengths of sides AB, BC, and AC using vector subtraction and the Pythagorean theorem.

Highlight: Vector operations provide a powerful tool for analyzing geometric properties of shapes in 3D space.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Planes and Coordinate Systems

This page delves into the representation of planes in 3D space and their relationship to coordinate systems. It explains how to determine points that lie on a given plane.

Definition: A plane in 3D space can be represented by an equation in the form x = a + ru + sv, where a is a point on the plane, and u and v are direction vectors.

The page provides examples of how to:

  1. Determine two points that lie on a given plane
  2. Represent planes using different coordinate axes

Highlight: Understanding plane equations is essential for solving problems involving the intersection of planes and lines.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Line-Plane Relationships

This page focuses on determining whether a point lies on a given line and explores the relationships between lines and planes in 3D space.

Example: The page demonstrates how to check if the point (2,3,-1) lies on the line g: x = (7,5,4) + t(-3,-5,6).

The solution involves substituting the point coordinates into the line equation and solving for the parameter t.

Vocabulary:

  • Durchstoßpunkt (piercing point): The point where a line intersects a plane

The page also introduces three possible relationships between a line and a plane:

  1. The line is parallel to the plane
  2. The line lies entirely within the plane
  3. The line intersects the plane at a single point
Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Line-Plane Intersection

This page continues the discussion of line-plane relationships, focusing on the case where a line lies entirely within a plane. It demonstrates how to determine this relationship mathematically.

Example: The page shows how to set up and solve a system of equations to check if a given line lies within a plane.

The solution involves equating the line and plane equations and analyzing the resulting system of equations.

Highlight: When a line lies within a plane, the system of equations has infinitely many solutions.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Coordinate Axis Intersections with a Plane

This page presents a problem involving finding the intersection points of a plane with the coordinate axes and determining if these points form an isosceles triangle.

Example: Given the plane E: x = (-8,5,6) + r(8,5,-9) + s(4,-5,1), find its intersections with the x, y, and z axes.

The solution involves setting up and solving systems of equations for each axis intersection.

Highlight: This problem combines concepts of plane equations, line-plane intersections, and triangle properties.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Continuation of Coordinate Axis Intersections

This page continues the solution from the previous page, completing the calculations for finding the intersection points of the plane with the coordinate axes.

The intersection points are found to be:

  • x-axis: (3,0,0)
  • y-axis: (0,3,0)
  • z-axis: (0,0,3)

Highlight: The symmetry of these intersection points (3 units along each axis) is noteworthy and simplifies further calculations.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Triangle Analysis

This page concludes the problem from the previous two pages by analyzing whether the triangle formed by the intersection points is isosceles.

The distances between the intersection points are calculated:

  • |AB| = √34
  • |BC| = √34
  • |AC| = √18

Conclusion: The triangle is not isosceles, as two sides have equal length (√34), but the third side has a different length (√18).

Highlight: This example demonstrates how vector and plane concepts can be applied to solve complex geometric problems.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Line Relationships in 3D Space

This page provides an overview of the possible relationships between lines in three-dimensional space. It introduces key concepts and terminology for understanding these relationships.

Vocabulary:

  • Schnittpunkt (intersection point): Where two lines meet
  • Windschief (skew): Lines that do not intersect and are not parallel

The page outlines four possible relationships between lines:

  1. Intersecting at a point
  2. Identical (completely overlapping)
  3. Parallel
  4. Skew

Highlight: Understanding these relationships is crucial for solving problems involving multiple lines in 3D space.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Öffnen

Determining Line Intersections

This page demonstrates how to determine if two lines intersect and find their point of intersection if they do. It provides a step-by-step example of the process.

Example: Given two lines, g₁ and g₂, the page shows how to set up and solve a system of equations to find their intersection point.

The solution involves:

  1. Equating the parametric equations of the two lines
  2. Solving the resulting system of equations
  3. Checking if the solution represents a valid intersection point

Highlight: The intersection point is found to be (3,1,6), demonstrating a successful application of the method.

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Gauß-Verfahren einfach erklärt: Beispiele, Übungen und Lösungen

user profile picture

brn01

@brn01

·

35 Follower

Follow

The Gauß-Verfahren einfach erklärt and geometric relationships between lines and planes form the core of analytical geometry. This comprehensive guide covers solving linear equation systems, vector calculations, and analyzing positional relationships between geometric objects in three-dimensional space.

  • The Gauß-Verfahren Matrix is demonstrated through detailed examples showing step-by-step solutions
  • Vector operations include calculating Richtungsvektor und Stützvektor for lines and planes
  • Lagebeziehung Gerade Ebene concepts are explored through various scenarios including parallel, intersecting, and skew lines
  • Practical examples demonstrate how to determine if points lie on lines or planes
  • Special cases of Gauß-Verfahren unendlich viele Lösungen are explained with clear illustrations

19.10.2021

6030

 

12

 

Mathe

91

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Vector Representation and Geometric Relationships

This page focuses on vector representation and geometric relationships in three-dimensional space. It introduces key concepts for understanding spatial relationships between points, lines, and planes.

Vocabulary:

  • Stützvektor (support vector): A vector that defines a point on a line or plane
  • Richtungsvektor (direction vector): A vector that indicates the direction of a line or plane

The page includes a diagram showing various points in 3D space, labeled from A to H, which helps visualize the spatial relationships discussed.

Highlight: Understanding the representation of vectors and points in 3D space is crucial for solving problems involving lines and planes.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Vector Operations and Triangle Properties

This page explores vector operations and their application in determining geometric properties of triangles. It presents a specific example of using vectors to check if a triangle is isosceles.

Example: Given points A(1,2,3), B(2,4,3), and C(3,1,3), the page demonstrates how to calculate vector lengths to determine if the triangle is isosceles.

The solution involves calculating the lengths of sides AB, BC, and AC using vector subtraction and the Pythagorean theorem.

Highlight: Vector operations provide a powerful tool for analyzing geometric properties of shapes in 3D space.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Planes and Coordinate Systems

This page delves into the representation of planes in 3D space and their relationship to coordinate systems. It explains how to determine points that lie on a given plane.

Definition: A plane in 3D space can be represented by an equation in the form x = a + ru + sv, where a is a point on the plane, and u and v are direction vectors.

The page provides examples of how to:

  1. Determine two points that lie on a given plane
  2. Represent planes using different coordinate axes

Highlight: Understanding plane equations is essential for solving problems involving the intersection of planes and lines.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Line-Plane Relationships

This page focuses on determining whether a point lies on a given line and explores the relationships between lines and planes in 3D space.

Example: The page demonstrates how to check if the point (2,3,-1) lies on the line g: x = (7,5,4) + t(-3,-5,6).

The solution involves substituting the point coordinates into the line equation and solving for the parameter t.

Vocabulary:

  • Durchstoßpunkt (piercing point): The point where a line intersects a plane

The page also introduces three possible relationships between a line and a plane:

  1. The line is parallel to the plane
  2. The line lies entirely within the plane
  3. The line intersects the plane at a single point
Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Line-Plane Intersection

This page continues the discussion of line-plane relationships, focusing on the case where a line lies entirely within a plane. It demonstrates how to determine this relationship mathematically.

Example: The page shows how to set up and solve a system of equations to check if a given line lies within a plane.

The solution involves equating the line and plane equations and analyzing the resulting system of equations.

Highlight: When a line lies within a plane, the system of equations has infinitely many solutions.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Coordinate Axis Intersections with a Plane

This page presents a problem involving finding the intersection points of a plane with the coordinate axes and determining if these points form an isosceles triangle.

Example: Given the plane E: x = (-8,5,6) + r(8,5,-9) + s(4,-5,1), find its intersections with the x, y, and z axes.

The solution involves setting up and solving systems of equations for each axis intersection.

Highlight: This problem combines concepts of plane equations, line-plane intersections, and triangle properties.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Continuation of Coordinate Axis Intersections

This page continues the solution from the previous page, completing the calculations for finding the intersection points of the plane with the coordinate axes.

The intersection points are found to be:

  • x-axis: (3,0,0)
  • y-axis: (0,3,0)
  • z-axis: (0,0,3)

Highlight: The symmetry of these intersection points (3 units along each axis) is noteworthy and simplifies further calculations.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Triangle Analysis

This page concludes the problem from the previous two pages by analyzing whether the triangle formed by the intersection points is isosceles.

The distances between the intersection points are calculated:

  • |AB| = √34
  • |BC| = √34
  • |AC| = √18

Conclusion: The triangle is not isosceles, as two sides have equal length (√34), but the third side has a different length (√18).

Highlight: This example demonstrates how vector and plane concepts can be applied to solve complex geometric problems.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Line Relationships in 3D Space

This page provides an overview of the possible relationships between lines in three-dimensional space. It introduces key concepts and terminology for understanding these relationships.

Vocabulary:

  • Schnittpunkt (intersection point): Where two lines meet
  • Windschief (skew): Lines that do not intersect and are not parallel

The page outlines four possible relationships between lines:

  1. Intersecting at a point
  2. Identical (completely overlapping)
  3. Parallel
  4. Skew

Highlight: Understanding these relationships is crucial for solving problems involving multiple lines in 3D space.

Beispiel 1: GAU SVERFAHREN 2x1-3xz-5xs =4 2x₂ + x3 = 0 3x3 = 6 2x -3x₂-5-2=-1 2x2 + 2 = O x3 = 2 21-3-(-1)-10 = -A x2 x3 -2 x₁ = 1 x2 = -1 *

Determining Line Intersections

This page demonstrates how to determine if two lines intersect and find their point of intersection if they do. It provides a step-by-step example of the process.

Example: Given two lines, g₁ and g₂, the page shows how to set up and solve a system of equations to find their intersection point.

The solution involves:

  1. Equating the parametric equations of the two lines
  2. Solving the resulting system of equations
  3. Checking if the solution represents a valid intersection point

Highlight: The intersection point is found to be (3,1,6), demonstrating a successful application of the method.

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Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Ranked #1 Education App

Laden im

Google Play

Laden im

App Store

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

Durchschnittliche App-Bewertung

15 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 12 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.