Chapter Two
Vectors-Algebra and Geometry
2.1 Vectors A directed line segment in space is a line segment together with a direction. Thus the directed line segment from the point P to the point Q is different from the directed line segment from Q to P. We f
Chapter 4
VECTORS AND
FOUNDATIONS
1
Algebraic operations with vectors
119. Denition of vectors. In physics, some quantities (e.g.
distances, volumes, temperatures, or masses) are completely characterized by their magnitudes, expressed with respect to a ch
Calculus II (part 6a): Vectors and 3D Geometry
1
(by Evan Dummit, 2012, v. 1.50)
Vectors and 3-Dimensional Geometry
We are used to graphing curves in the plane given a function
y = f (x).
y = 1 + cos(t).
x2 + y 2 = 1.
curve
f (x), we know how to produce t
1
CHAPTERONE
VECTORGEOMETRY
1.1INTRODUCTION
Inthischaptervectorsarefirstintroducedasgeometricobjects,namelyasdirectedline
segments,orarrows. Theoperationsofaddition,subtraction,andmultiplicationbya
scalar (real number) are defined for these directed line
Vector Geometry
What is a vector?
Definition: A directed line segment
Characteristics:
Tail (initial point)
Tip (terminal point)
Vectors vs. Scalars
A vector has:
A vector is used to represent quantities like:
A scalar has:
A scalar is used to represent q
Vectors and Plane Geometry
Karl Heinz Dovermann
Professor of Mathematics
University of Hawaii
January 27, 2011
Preface
During the rst week of the semester it is dicult to get started with the
course material. Some students have not settled in, some are st
Vector Geometry Final Exam Review
4x
4y
Problem 1. Find the center and the radius for the sphere x2 +
+ y2 + z2
z 7 = 0. Note: Recall
3
3
that the center and the radius of a sphere
x2 + ax + y 2 + by + z 2 = d
are
ab
c
, ,
222
and
d+
a2
b2
c2
+
+ , res
314 CHAPTER 12 I VECTOR GEOMETRV (LT CHAPTER 13)
4. Let v = Q, where P = (1. 1) and Q = (2. 2). What is the head of the vector v' equivalent to v based at (2, 4)?
What is the head of the vector v0 equivalent to v based at the origin? Sketch v, v0, and
Chapter 1
Vector Algebra
1.1
Rectangular Coordinates in 1-Space, 2-Space and
3-Space
The Euclidean Geometry is the study of geometric elements (points, lines,
planes, etc), relations between elements and congurations. The simplest
geometric elements and b
July 17, 2008 Version
Free to photocopy and distribute
David A. SANTOS
[email protected]
Mathesis iuvenes tentare rerum quaelibet ardua semitasque non usitatas pandere docet.
ii
Copyright c 2007 David Anthony SANTOS. Permission is granted to copy, distr
Chapter 4
Introduction to Vector Geometry
4.1
Vectors in the Plane
The 2-dimensional plane is described by a pair of axes labelled X and Y . Each point in the
plane corresponds to the ordered pair of real numbers, consisting rstly of its X -coordinate and
I. Vectors and Geometry in Two and Three Dimensions
I.1 Points and Vectors
Each point in two dimensions may be labeled by two coordinates (a, b) which specify the position of
the point in some units with respect to some axes as in the gure on the left bel