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Geometria solida

Geometria nello Spazio: Equazioni e Distanze

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Geometria nello Spazio: Equazioni e Distanze

Irene

@irene_mapp

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The document provides a comprehensive overview of analytic geometry in three-dimensional space, covering key concepts like coordinate systems, planes, vectors, and lines. It explains how to represent points, calculate distances, and determine equations for geometric objects in 3D space.

Key topics include:
• Cartesian coordinates in 3D space
• Equations of planes and lines
• Vector operations and properties
• Relative positions of planes and lines
• Distance calculations
• Parametric and Cartesian equations

The material progresses from basic definitions to more complex problem-solving involving multiple geometric objects. Numerous examples demonstrate practical applications of the concepts.

21/2/2023

5548

Vector Operations in 3D Space

This section introduces fundamental vector operations and concepts in three-dimensional space. It begins by defining a versore as a unit vector with direction and orientation corresponding to the Cartesian axes.

Key vector operations covered include:

Vector Addition
Vector Subtraction
Scalar Multiplication
Dot Product

Definition: The dot product of two vectors v1 and v2 is defined as: v1 • v2 = |v1| |v2| cos θ where θ is the angle between the vectors.

The section also introduces important vector relationships:

Definition: Parallel vectors v and w satisfy the condition: v = kw, where k is a scalar

Definition: Perpendicular vectors v and w satisfy the condition: v • w = vxwx + vywy + vz*wz = 0

These operations and relationships form the foundation for more complex geometric analyses in 3D space, particularly in determining the relative positions of planes and lines.

A
geometria s
analitica
COORDINATE CARTESIANE NELLO SPAZIO
Un punto e' dato da una terna ordinata di numeri che
rappresentano le coordinate

Relative Positions of Planes

This section explores the possible relative positions of two planes in 3D space. Given two planes:

π1: ax + by + cz + d = 0 π2: a'x + b'y + c'z + d' = 0

Three possible configurations are discussed:

Parallel Distinct Planes
Parallel Coincident Planes
Intersecting Planes

For parallel planes, the condition a/a' = b/b' = c/c' must be satisfied. If this condition is met and d/d' is also equal to the same ratio, the planes are coincident.

Intersecting planes are characterized by normal vectors that are not parallel. The intersection of two planes forms a line.

Highlight: Intersecting planes can be perpendicular, satisfying the condition: n1 • n2 = 0, or aa' + bb' + cc' = 0

An example is provided to illustrate how to determine the relative position of two given planes:

Example: For planes π1: 3x + y - z - 1 = 0 and π2: 6x + 2y - 2z + 2 = 0, it is shown that they are parallel and distinct.

The section concludes with another example demonstrating how to find the equation of a plane passing through two points and parallel to a given axis, showcasing the practical application of these concepts.

Vedi

Cartesian Coordinates in 3D Space

This section introduces the fundamental concept of representing points in three-dimensional space using Cartesian coordinates. A point P is defined by an ordered triple (x, y, z) representing its position along the x, y, and z axes respectively.

The coordinate planes are defined: • x = 0 plane: yz plane • y = 0 plane: xz plane
• z = 0 plane: xy plane (the familiar 2D plane)

Important formulas are provided for working with points in 3D space:

Definition: Distance between two points A(xA, yA, zA) and B(xB, yB, zB): AB = √[(xA-xB)² + (yA-yB)² + (zA-zB)²]

Definition: Midpoint M between two points A and B: M = ((xA+xB)/2, (yA+yB)/2, (zA+zB)/2)

An example is given to illustrate these concepts, involving finding the length and midpoint of a line segment between two given points.

Example: For points A(1,2,4) and B(0,3,-2), the distance AB ≈ 6.16 and the midpoint M is (1/2, 5/2, 1).

Vedi

The Line in 3D Space

This section delves into the various ways of representing a line in three-dimensional space. The equation of a line can be expressed in multiple forms:

Parametric Equations: x = x0 + ke y = y0 + km z = z0 + kn where (e, m, n) is the direction vector and (x0, y0, z0) is a point on the line.
Cartesian Equations: (x - x0)/e = (y - y0)/m = (z - z0)/n = k
Line Passing Through Two Points: For points A(x1, y1, z1) and B(x2, y2, z2): (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1) = k
Intersection of Two Intersecting Planes: The line can be represented as the solution to the system of equations of two intersecting planes.

Example: The section provides an example of determining the equation of a line passing through points A(-1, 4, -5) and B(1, 2, -1).

Another example demonstrates how to find both parametric and Cartesian equations for a line passing through a given point and having a specified direction vector.

These various representations of lines in 3D space are fundamental in solving complex geometric problems and are widely used in fields such as computer graphics, robotics, and engineering design.

Vedi

Distance from a Point to a Plane

This section introduces the formula for calculating the distance between a point and a plane in three-dimensional space.

Definition: Given a point P(x0, y0, z0) and a plane π: ax + by + cz + d = 0, the distance is calculated as:

d(P,π) = |ax0 + by0 + cz0 + d| / √(a² + b² + c²)

This formula is crucial for various applications in 3D geometry, including determining the shortest distance between objects and analyzing spatial relationships.

Example: For the plane π: 3x + y - z = 0 and the point P(-2, 4, 1), the distance is calculated to be 3√11 / 11.

The section provides a step-by-step solution to this example, demonstrating how to apply the formula in practice. This calculation is essential in many real-world applications, such as computer graphics, robotics, and structural engineering.

Vedi

The Plane in 3D Space

This section delves into the properties and equations of planes in three-dimensional space. The general equation of a plane π is introduced as:

ax + by + cz + d = 0

Where the vector n(a,b,c) is defined as the normal vector to the plane, being perpendicular to it.

Two methods for determining the equation of a plane are presented:

Given three points
Given a point and the normal vector

For the first method, the process involves substituting the coordinates of the three given points into the general equation, resulting in a system of three equations with four unknowns (a, b, c, d). Three parameters are then expressed in terms of the fourth, which is called the free parameter and can be eliminated as it is non-zero.

Example: For points A(2,0,0), B(0,1,0), and C(0,0,3), the resulting plane equation is 3x + 6y + 2z - 6 = 0.

For the second method, the components of the normal vector are directly used as coefficients a, b, and c in the general equation. The d term is then determined by imposing the condition that the plane passes through the given point.

Example: For point A(3,5,0) and normal vector n(0,1,0), the resulting plane equation is y - 5 = 0.

These methods provide a comprehensive approach to defining planes in 3D space, which is crucial for more complex geometric analyses.

Vedi

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Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

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Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

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Adoro questa app ❤️, la uso praticamente sempre quando studio.

Non c'è niente di adatto? Esplorare altre aree tematiche.

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 11 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS

Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Geometria

Geometria solida

Geometria nello Spazio: Equazioni e Distanze

Irene

@irene_mapp

12 Follower

Segui

The material progresses from basic definitions to more complex problem-solving involving multiple geometric objects. Numerous examples demonstrate practical applications of the concepts.

21/2/2023

5548

4ªl

Matematica

233

Vector Operations in 3D Space

Key vector operations covered include:

Vector Addition
Vector Subtraction
Scalar Multiplication
Dot Product

Definition: The dot product of two vectors v1 and v2 is defined as: v1 • v2 = |v1| |v2| cos θ where θ is the angle between the vectors.

The section also introduces important vector relationships:

Definition: Parallel vectors v and w satisfy the condition: v = kw, where k is a scalar

Definition: Perpendicular vectors v and w satisfy the condition: v • w = vxwx + vywy + vz*wz = 0

These operations and relationships form the foundation for more complex geometric analyses in 3D space, particularly in determining the relative positions of planes and lines.

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Relative Positions of Planes

This section explores the possible relative positions of two planes in 3D space. Given two planes:

π1: ax + by + cz + d = 0 π2: a'x + b'y + c'z + d' = 0

Three possible configurations are discussed:

Parallel Distinct Planes
Parallel Coincident Planes
Intersecting Planes

For parallel planes, the condition a/a' = b/b' = c/c' must be satisfied. If this condition is met and d/d' is also equal to the same ratio, the planes are coincident.

Intersecting planes are characterized by normal vectors that are not parallel. The intersection of two planes forms a line.

Highlight: Intersecting planes can be perpendicular, satisfying the condition: n1 • n2 = 0, or aa' + bb' + cc' = 0

An example is provided to illustrate how to determine the relative position of two given planes:

Example: For planes π1: 3x + y - z - 1 = 0 and π2: 6x + 2y - 2z + 2 = 0, it is shown that they are parallel and distinct.

Iscriviti per mostrare il contenuto. È gratis!

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Cartesian Coordinates in 3D Space

The coordinate planes are defined: • x = 0 plane: yz plane • y = 0 plane: xz plane
• z = 0 plane: xy plane (the familiar 2D plane)

Important formulas are provided for working with points in 3D space:

Definition: Distance between two points A(xA, yA, zA) and B(xB, yB, zB): AB = √[(xA-xB)² + (yA-yB)² + (zA-zB)²]

Definition: Midpoint M between two points A and B: M = ((xA+xB)/2, (yA+yB)/2, (zA+zB)/2)

An example is given to illustrate these concepts, involving finding the length and midpoint of a line segment between two given points.

Example: For points A(1,2,4) and B(0,3,-2), the distance AB ≈ 6.16 and the midpoint M is (1/2, 5/2, 1).

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The Line in 3D Space

This section delves into the various ways of representing a line in three-dimensional space. The equation of a line can be expressed in multiple forms:

Parametric Equations: x = x0 + ke y = y0 + km z = z0 + kn where (e, m, n) is the direction vector and (x0, y0, z0) is a point on the line.
Cartesian Equations: (x - x0)/e = (y - y0)/m = (z - z0)/n = k
Line Passing Through Two Points: For points A(x1, y1, z1) and B(x2, y2, z2): (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1) = k
Intersection of Two Intersecting Planes: The line can be represented as the solution to the system of equations of two intersecting planes.

Example: The section provides an example of determining the equation of a line passing through points A(-1, 4, -5) and B(1, 2, -1).

Another example demonstrates how to find both parametric and Cartesian equations for a line passing through a given point and having a specified direction vector.

These various representations of lines in 3D space are fundamental in solving complex geometric problems and are widely used in fields such as computer graphics, robotics, and engineering design.

Iscriviti per mostrare il contenuto. È gratis!

Accesso a tutti i documenti

Unisciti a milioni di studenti

Migliora i tuoi voti

Iscrivendosi si accettano i Termini di servizio e la Informativa sulla privacy.

Distance from a Point to a Plane

This section introduces the formula for calculating the distance between a point and a plane in three-dimensional space.

Definition: Given a point P(x0, y0, z0) and a plane π: ax + by + cz + d = 0, the distance is calculated as:

d(P,π) = |ax0 + by0 + cz0 + d| / √(a² + b² + c²)

This formula is crucial for various applications in 3D geometry, including determining the shortest distance between objects and analyzing spatial relationships.