Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Vector Parametrizations.

Slides:

Advertisements

Similar presentations

Volume by Parallel Cross Section; Disks and Washers

Advertisements

Euclidean m-Space & Linear Equations Euclidean m-space.

MA Day 9 – January 17, 2013 Review: Equations of lines, Section 9.5 Section 9.5 –Planes.

Rubono Setiawan, S.Si.,M.Sc.

Chapter 12 – Vectors and the Geometry of Space

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Consider.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form.

VECTORS AND THE GEOMETRY OF SPACE 12. VECTORS AND THE GEOMETRY OF SPACE A line in the xy-plane is determined when a point on the line and the direction.

Lines and Planes in Space

11 Analytic Geometry in Three Dimensions

Vectors: planes. The plane Normal equation of the plane.

Copyright © Cengage Learning. All rights reserved. 10 Analytic Geometry in Three Dimensions.

Section 9.5: Equations of Lines and Planes

Assigned work: pg. 449 #1abc, 2abc, 5, 7-9,12,15 Write out what “you think” the vector, parametric a symmetric equations of a line in R 3 would be.

Vectors and the Geometry of Space

Eqaution of Lines and Planes.  Determine the vector and parametric equations of a line that contains the point (2,1) and (-3,5).

Assigned work: pg. 433 #1-12 Equation of a line – slope and point or two points BUT NOW we will learn to describe an Equation of a Line by using vectors…………………..

10.5 Lines and Planes in Space Parametric Equations for a line in space Linear equation for plane in space Sketching planes given equations Finding distance.

Chapter 9-Vectors Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Double Integrals a. (16.2.1),

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom.

The Vector Product. Remember that the scalar product is a number, not a vector Using the vector product to multiply two vectors, will leave the answer.

Assigned work: pg. 468 #3-8,9c,10,11,13-15 Any vector perpendicular to a plane is a “normal ” to the plane. It can be found by the Cross product of any.

Properties of Vector Operations: u, v, w are vectors. a, b are scalars. 0 is the zero vector. 0 is a scalar zero. 1. u + v = v + u 2. (u + v) + w = u +

Copyright © Cengage Learning. All rights reserved. 12 Vectors and the Geometry of Space.

Section 3.5 Lines and Planes in 3-Space. Let n = (a, b, c) ≠ 0 be a vector normal (perpendicular) to the plane containing the point P 0 (x 0, y 0, z 0.

1.3 Lines and Planes. To determine a line L, we need a point P(x 1,y 1,z 1 ) on L and a direction vector for the line L. The parametric equations of a.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Elementary Examples a.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Chapter 12: Vectors Cartesian.

Equations of Lines and Planes

1. Given vectors a, b, and c: Graph: a – b + 2c and 3c – 2a + b 2. Prove that these following vectors a = 3i – 2j + k, b = i – 3j +5k, and c = 2i +j –

Functions of Several Variables Copyright © Cengage Learning. All rights reserved.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Derivative a. Tangent.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Line Integrals a. Definition.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved (p. 443) First Area.

Lines and Planes In three dimensions, we use vectors to indicate the direction of a line. as a direction vector would indicate that Δx = 7, Δy = 6, and.

VECTORS AND THE GEOMETRY OF SPACE 12. PLANES Thus, a plane in space is determined by:  A point P 0 (x 0, y 0, z 0 ) in the plane  A vector n that is.

DOT PRODUCT CROSS PRODUCT APPLICATIONS

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Vector Functions a. Vector.

CHAPTER 10 DATA COLLECTION METHODS. FROM CHAPTER 10 Copyright © 2003 John Wiley & Sons, Inc. Sekaran/RESEARCH 4E.

homogeneous coordinates equationmisc point(w ; x, y, z)r w = S 0 where S 0 = xi + yj + zk  3 points in 3D space.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Bernoulli Equations: Homogeneous.

Parametric and general equation of a geometrical object O EQUATION INPUT t OUTPUT EQUATION INPUT OUTPUT (x, y, z)

3D Lines and Planes.

Extended Work on 3D Lines and Planes. Intersection of a Line and a Plane Find the point of intersection between the line and the plane Answer: (2, -3,

Lines and Planes In three dimensions, we use vectors to indicate the direction of a line. as a direction vector would indicate that Δx = 7, Δy = 6, and.

Chapter 18: Line Integrals and Surface Integrals

Chapter 3: Differentiation Topics

Chapter 14: Vector Calculus

Copyright © Cengage Learning. All rights reserved.

Chapter 3 VECTORS.

VECTORS APPLICATIONS NHAA/IMK/UNIMAP.

11 Vectors and the Geometry of Space

Lines and Planes in Space

13 Functions of Several Variables

Double Integrals We start with a function f continuous on a rectangle

Notation of lines in space

Find a vector equation for the line through the points {image} and {image} {image}

Find a vector equation for the line through the points {image} and {image} {image}

By the end of Week 3: You would learn how to solve many problems involving lines/planes and manipulate with different coordinate systems. These are.

Vectors and the Geometry

Vectors and the Geometry

Lesson 83 Equations of Planes.

Double Integrals We start with a function f continuous on a rectangle

Copyright © Cengage Learning. All rights reserved.

11 Vectors and the Geometry of Space

Chapter 17: Line Integrals and Surface Integrals

Chapter 16: Double and Triple Integrals

Vectors and the Geometry

Power Series Salas, Hille, Etgen Calculus: One and Several Variables

Presentation transcript:

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Vector Parametrizations

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Parametrize the line that passes through the point P (1,-1,2) and has direction vector 2i-3j+k

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Scalar Parametric Equations Symmetric Form

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Write a parametric equations for the line that passes through the point P (1,-1,2) and has direction vector 2i-3j+k

Main Menu Intersecting Lines, Parallel Lines Two distinct lines l 1 : r(t) = r 0 + td, l 2 : R(u) = R 0 + uD intersect iff there are numbers t and u at which r(t) = R(u). Example: Find the point at which the lines Intersect.

Main Menu Example: (1) Find the angle between the lines (2) Find the parametrization for the line that passes through their intersection and is perpendicular to both l 1 and l 2.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Distance from a Point to a Line Let P 0 be a point on l and let d be a direction vector for l. With P 0 and Q as shown in the figure, you can see that

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Scalar Equation of a Plane

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Vector Equation of a Plane We can write the equation of a plane entirely in vector notation. Set Since r 0 = x 0 i + y 0 j + z 0 k and r = xi + yj + zk, (13.6.1) can be written

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Intersecting Planes Unit Normals If N is normal to a given plane, then all other normals to that plane are parallel to N and hence scalar multiples of N. In particular there are two normals of length 1:

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Show that the plane are non-parallel and find a scalar parametric equations for the line formed by the two intersecting planes.

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes The Plane Determined by Three Noncollinear Points

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes The Distance from a Point to a Plane