Inner Products, Length, and Orthogonality (11/30/05) If v and w are vectors in R n, then their inner (or dot) product is: v  w = v T w That is, you multiply.

Slides:

Advertisements

Similar presentations

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

Advertisements

Vectors Lesson 4.3.

Section 9.3 The Dot Product

6 6.1 © 2012 Pearson Education, Inc. Orthogonality and Least Squares INNER PRODUCT, LENGTH, AND ORTHOGONALITY.

Vector Spaces (10/27/04) The spaces R n which we have been studying are examples of mathematical objects which have addition and scalar multiplication.

Vectors and Vector Equations (9/14/05) A vector (for us, for now) is a list of real numbers, usually written vertically as a column. Geometrically, it’s.

6 6.1 © 2012 Pearson Education, Inc. Orthogonality and Least Squares INNER PRODUCT, LENGTH, AND ORTHOGONALITY.

Orthogonal Sets (12/2/05) Recall that “orthogonal” matches the geometric idea of “perpendicular”. Definition. A set of vectors u 1,u 2,…,u p in R n is.

Chapter 14 Section 14.3 Curves. x y z To get the equation of the line we need to know two things, a direction vector d and a point on the line P. To find.

VECTORS AND THE GEOMETRY OF SPACE 12. VECTORS AND THE GEOMETRY OF SPACE So far, we have added two vectors and multiplied a vector by a scalar.

6.4 Vectors and Dot Products

Length and Dot Product in R n Notes: is called a unit vector. Notes: The length of a vector is also called its norm. Chapter 5 Inner Product Spaces.

Section 9.5: Equations of Lines and Planes

The Cross Product Third Type of Multiplying Vectors.

CS 3388 Working with 3D Vectors

Vectors and the Geometry of Space

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

REVIEW OF MATHEMATICS. Review of Vectors Analysis GivenMagnitude of vector: Example: Dot product:  is the angle between the two vectors. Example:

Chapter 5 Orthogonality.

Angles Between Vectors Orthogonal Vectors

Chapter 3 Vectors in n-space Norm, Dot Product, and Distance in n-space Orthogonality.

6 6.1 © 2016 Pearson Education, Inc. Orthogonality and Least Squares INNER PRODUCT, LENGTH, AND ORTHOGONALITY.

Assigned work: pg.377 #5,6ef,7ef,8,9,11,12,15,16 Pg. 385#2,3,6cd,8,9-15 So far we have : Multiplied vectors by a scalar Added vectors Today we will: Multiply.

Chapter 10 Real Inner Products and Least-Square

Honors Pre-Calculus 12-4 The Dot Product Page: 441 Objective: To define and apply the dot product.

Basic Entities Scalars - real numbers sizes/lengths/angles Vectors - typically 2D, 3D, 4D directions Points - typically 2D, 3D, 4D locations Basic Geometry.

A rule that combines two vectors to produce a scalar.

Section 3.5 Revised ©2012 |

Inner Product, Length and Orthogonality Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

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

Section 6.2 Angles and Orthogonality in Inner Product Spaces.

Discrete Math Section 12.4 Define and apply the dot product of vectors Consider the vector equations; (x,y) = (1,4) + t its slope is 3/2 (x,y) = (-2,5)

Section 4.2 – The Dot Product. The Dot Product (inner product) where is the angle between the two vectors we refer to the vectors as ORTHOGONAL.

Section 3.3 Dot Product; Projections. THE DOT PRODUCT If u and v are vectors in 2- or 3-space and θ is the angle between u and v, then the dot product.

Section 9.3: The Dot Product Practice HW from Stewart Textbook (not to hand in) p. 655 # 3-8, 11, 13-15, 17,

Chapter 4 Vector Spaces Linear Algebra. Ch04_2 Definition 1: ……………………………………………………………………. The elements in R n called …………. 4.1 The vector Space R n Addition.

6.4 Vector and Dot Products. Dot Product  This vector product results in a scalar  Example 1: Find the dot product.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Basics Normalizing a vector => unit vector Dot product Cross product Reflection vector Parametric form.

The Matrix Equation A x = b (9/16/05) Definition. If A is an m  n matrix with columns a 1, a 2,…, a n and x is a vector in R n, then the product of A.

Precalculus Section 14.1 Add and subtract matrices Often a set of data is arranged in a table form A matrix is a rectangular.

Objectives: Graph vectors in 3 dimensions Use cross products and dot products to find perpendicular vectors.

An inner product on a vector space V is a function that, to each pair of vectors u and v in V, associates a real number and satisfies the following.

We will use the distance formula and the law of cosines to develop a formula to find the angle between two vectors.

Dr. Shildneck. Dot Product Again, there are two types of Vector Multiplication. The inner product,called the Dot Product, and the outer product called.

Chapter 4 Vector Spaces Linear Algebra. Ch04_2 Definition 1. Let be a sequence of n real numbers. The set of all such sequences is called n-space (or.

6 6.1 © 2016 Pearson Education, Ltd. Orthogonality and Least Squares INNER PRODUCT, LENGTH, AND ORTHOGONALITY.

Math /7.5 – Vectors 1. Suppose a car is heading NE (northeast) at 60 mph. We can use a vector to help draw a picture (see right). 2.

CSE 681 Brief Review: Vectors. CSE 681 Vectors Direction in space Normalizing a vector => unit vector Dot product Cross product Parametric form of a line.

1 Chapter 6 Orthogonality 6.1 Inner product, length, and orthogonality 内积，模长，正交 6.2 Orthogonal sets 正交组，正交集 6.4 The Gram-Schmidt process 格莱姆 - 施密特过程.

Vectors Def. A vector is a quantity that has both magnitude and direction. v is displacement vector from A to B A is the initial point, B is the terminal.

Dot Product of Vectors.

Vector projections (resolutes)

Matrix Arithmetic Prepared by Vince Zaccone

Dot Product and Angle Between Two Vectors

Elementary Linear Algebra

Vectors and Angles Lesson 10.3b.

Scalars and Vectors.

Law of sines Law of cosines Page 326, Textbook section 6.1

Angles Between Vectors Orthogonal Vectors

Scalar and Vector Projections

Inner Product, Length and Orthogonality

Section 3.2 – The Dot Product

12.3 The Dot Product.

10.6: Applications of Vectors in the Plane

Homework Questions!.

Linear Algebra Lecture 38.

Enumerations, Clamping, Vectors

CHAPTER 3 VECTORS NHAA/IMK/UNIMAP.

Math with Vectors I. Math Operations with Vectors.

Vectors Lesson 4.3.

Presentation transcript:

Inner Products, Length, and Orthogonality (11/30/05) If v and w are vectors in R n, then their inner (or dot) product is: v  w = v T w That is, you multiply the corresponding entries and add up, so the result is a scalar. For example, (2,1,5)  (4,-3,1) = 10

Length (or Norm) The length or norm of a vector v is ||v|| =  (v  v) For example, ||(2,5,-1)|| =  30 A unit vector is a vector whose length is 1. Note that for any vector v, the vector v / ||v|| is a unit vector (it has been “normalized”).

Distance between two vectors The distance between two vectors v and w is just ||v – w|| For example, the distance between (2,5,-1) and (1,-3,-4) is ||(1,8,3)|| =  74.

Orthogonality Two vectors v and w are said to be orthogonal if v  w = 0. Orthogonality generalizes the idea of perpendicularity. A vector v is orthogonal to a subspace W if v is orthogonal to every vector in W. The set of all such vectors is called the orthogonal complement of W.

Assignment for Friday Read Section 6.1 carefully. Read these summarizing slides. On pages 382-3, do Exercises 1-19 odd and 27.