MATH1131 Lecture Notes - Lecture 5: Lithium Chloride, Orthogonality, Rishi
3 Jun 2018
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r
2 Vector Geo.rnetry
We have already seer that v@tors caa tre used very ef-
fectively for proving georretrie tesult*. Ia *his chapter
we continue studying vectors and we shall see how they
can be used to find shortwt distaaces between points,
lines and planes, as well as some other applieations. As
always, we need to le,rn soire new concepts first.
Recall that srtrier we defiaed the length of a vector a €
IR." to be' '
ir t
rl "-i
The reason fqg this dg&ifjoa y'qs-that if r'r : 2 or 3 this
--.t
is just Plthagora€' theorern and so,-it is {ritiuat to extend
the definit-ioutu'higher dtuieqsio.ns. * ,
We do a similar thing for the *gte Uett"*., 2 vectors,
but first we need a simple definition.
Proposition 2.3.
1. a-b-b.a.
2. a'(io') : .\(a'b).
3. a.(b * c) : a'h + a'c'
4. (afb).":a.c*b.c.
As a congeqrrcnce
(a+b)-(e +d): *,*.*{'d, *
3.1 The do't pr6duct.
Dssnition 2.!. f,et a, b € Rt. Tke da* produet o/
these twt, uecta:rs is
.:R*xlRor+lR
"r I
Proposition 2.2. ., ' .: ,.
:la
' j , r'l i'
: . a'a-lal-.
H'e€S,'+[ifl ': :'
:,
,L, {'= a,qri al4: r . ..4 eaAo
' :..: " --, ,' A;r'+ dr. + , J,:. .t 4r.*
-lcl. . ' i{ r.'
LThis is a very use{i:l result, arrd wewi$ trse it oftea. Sor*e
other useful, and equally eany rwults to prove are:
; -t
r I .lr I
Example 2.4"
t'
Find a-b.
Leta: andb:
:
t
[1)
'.i
(i)
*, h., (i) (j)
-l*+-i-tL
-l2r
b.t*b,L
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