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Lecture

Vectors, projection

13 Pages
137 Views

Department
Mathematics
Course Code
MATA23H3
Professor
Sophie Chrysostomou

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VectorsintheEuclidean n-space
DEFINITION:Ifnisapositiveinteger,theEuclidean n-space,Rn,isthe collectionof
all ordered n-tuplesofrealnumbers.
Therearetwotypesofn-tuplesinRn:
(a1,a2,···,an)
[a1,a2,···,an]
ThezerovectorinRnisthevectorcontainingzeroesinall ofitscomponents:0=[0,0,···,0].
GeometricInterpretationofVectors.
InRInR2:InR3
www.notesolution.com
We cangeneralize and saythat thevectora=[a1,a2,···,an], inits standard position, is
thearrowthatstartsat theorigin(0,0,···,0)and endsat thepoint (a1,a2,···,an).
Avectorofthesamelengthand directionasa, isthevectoratranslatedto anotherposition
inRn.For thisreasonitisalsocalleda.
VectorAlgebrainRn:Letv=[v1,v2,···,vn]and w=[w1,w2,···,wn]beinRnand
rR.Wedefine
1.VectorAddition:v+w=[v1+w1,v2+w2,···,vn+wn].
2.VectorSubtraction:vw=[v1w1,v2w2,···,vnwn].
3.ScalarMultiplication:rv=[rv1,rv2,···,rvn].
EXAMPLE:GeometricInterpretation
2c
2011 bySophieChrysostomou
www.notesolution.com
DEFINITION:ParallelVectorsWesaythat twononzerovectorsvand wareparallel
and wedenoteitbyvkw, ifv=rwforsomerealnumberr.
Supposev=rw
0<r<1r>1
1<r<0r<1
3c
2011 bySophieChrysostomou
www.notesolution.com

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Description
Vectors in the Euclidean n-space n DEFINITION: If n is a positive integer, the Euclidean n-space, R , is the collection of all ordered n-tuples of real numbers. n There are two types of n-tuples in R : (a1,a2,,a n [a1,a2,,a n The zero vector in R is the vector containing zeroes in all of its components: 0= [0,0,,0]. Geometric Interpretation of Vectors. 2 3 In R In R : In R www.notesolution.comWe can generalize and say that the vector a = [a ,a ,1,a2], in ins standard position, is the arrow that starts at the origin (0,0,,0) and ends at the point (a ,a ,1,2 ) . n A vector of the same length and direction as a, is the vector a translated to another position in R . For this reason it is also called a. n n Vector Algebra in R : Let v = [v ,1 ,2,v ] nnd w = [w ,w ,1,w2] be innR and r R. We dene 1. Vector Addition: v + w = [v +1w ,v 1 w2,,2 + w ].n n 2. Vector Subtraction: v w = [v 1w ,v 1 w2,,2 w ].n n 3. Scalar Multiplication: rv = [rv 1rv ,2,rv ].n EXAMPLE: Geometric Interpretation 2 2011 by Sophie Chrysostomou www.notesolution.com
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