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Multivariable Calculus All in one - Chapter 1
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Multivariable Calculus All in one - Chapter 1, covering dot product, cross product, tangent line and plane and examples
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1.1 VectorsAvectoris anorderedpair in N 1⼼
triplet ㄑ 1，2
ntupknMNN M P
NN
⼼ 以 .Noi.TN
⼆ visualized
as n a positionvector
剾 uni a directedlinesegment
from the
ˋ 认成 tzwni.us originto lu.nl
⼆ 冷订 ⼗以下
For convenience we'dliketomovevectorsaround
It'sOKtoAtvectoraroundaslongas
notchangingdirect
onandmagnitude
i j representii.fithatstartsatpointA
ˋ
我 is thevectorfrompointAtop让 B
是 P.ATzebrabias is a for丽
representative
lgtavdrlormgnno
ˋ
lisdenotdbylilorMlordlAB
下 ⼼ 以 ⼼ bra bickbrass we .hr
侧 记 4vitvT fbia14hn define
analogyfor江 ⼼ 以 a. t's
g 下位 27 11圳WET222
i Notice
NIKOifandonlyif to ⼆⼼ 0
⼀⼀
the vector
zero
Hais arealnumber NkMM samedirectionasto
Po
NO apple
If the length1，11圳 that is called a unit forang vector itsunitvector is calculated
by
In 䢳 是以1，0，0 ⼩⼼ ⼼ 不⼼ 1 春⽉
g Findunited in thedirectionof Tzu 2，1 1.0.07 0，1，0 ⼗ o 0，1
i 订⼗下
幽 hide
tbgitnomlnomaket lltkhzi
fz.ro sothevectorwewantis 吉 li j.IN
唰䵺 嚙
, 加州
⽹ MI
Vector anonandsubtraction tea b nick bi t坑 a t b at b
E 古⽐11 古 在⼗万⼆万⼗⼤
i
a
愀
ǎs 亪 I 台⼗⼤汇
⼀
⼗万
吉 吉
f 记
cnn.tl
DotProduct
Let⼤⼼ 以 以7 古之 ⼼ 以 ⼼ dotpodudlinerpod.at and
Hiand元 is a realnumber
nth of t and is
t.fi UWithWatVsWs
Noticethat t.az v 以订阅
Relation
btwdotpdtandlengthit.tlHi so Ev 州1
Geometric version
ofdotpdt theanglebtw t andno is takentobethesmallerof two angles
so
OEQET .itparallel then GO
pointto oppositedirection not
if pointsame direction then Go
If t.tv arevectors n Nor IN then
i 㖄 㸠⼼ or
will
䬟 1Lawof an
㤺 瞓训训训训训㸠
定it wtilw tkit zt.tt 元元 11112⼀⼼到⼗INN
so 北州⼆州
训训吣
坑 北111㸠⽐
tando are orthogonalperpendicular y theanglebtwthenis E or900
Test
forwthgongk.tt⽆betwononzero vectors 讥 NorIN thenTando are orthogonal
if 坑
if 0涎 尤⽒20
if t.ro then cos0 0 0 E since以0㐳
1
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