Description
studySour
Functions of l variable
f(t)
input :
t
output : Jut)
Study Sour
vector functies
Single input  multiple outputs Multivariable functions
multiple inputs  multiple
outputs
Tools
Multi  dimentional space or the n dim. Space (nspace) R.
R  set of real nos also called the Real line . TR  plane (geenotasically); set of 2 ozdorly seal nos, or ozdared pairs
of real nos. (algebsically R'  3d space (Geometacale, set of ordered toi ples of seal nos R” – set of all ordered n  tubles oleval nos
Don't forget about the age old question of What does a molecular equation show?
(x,x ....... Xn) why is TR imp? or
eg NOA Ă has 1500 weather stations, each recording about 120 parameters per hour so every hour, we have 30,000 dimension We also discuss several other topics like What is the correct progression of the roman orders from the simplest to the most ornate?
variables to selve eg. Two points in n dimensions P (a,,,...., An) & Q (k,... bn)
Pol = N(a, b,j2+.....tanbn)?
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tip  We also discuss several other topics like What do people think of themselves?
tail
Splore of radius 'R' with center (a, a,...... and
Asphere consists of ponts P (X, *2,.... Xn) such that
(x,a 2 + (xnan) 2 = R2 For n=2, we estain egn of a circle If you want to learn more check out Do all states require a registered agent?
n=3, we obtain egn of a the 3D sphere, If you want to learn more check out Where was zeus born?
AB = <b, a,, ..... bnan) Addition of actores We also discuss several other topics like What is the difference between a democratic government and one that is authoritarian?
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Parallelogram dule
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I
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113
Iztea in R 2 vector at the same if Blb.,...be
tip AB = pa maantede a ductas AG, An) il cue can supozimfose one
on the other by PARALLEL
TRANSLATION The Ergin is defined as the point with all coordinates =0. 1 ,...)
OR is the position vector of Ŕ yo is the origin
cloaksically
ū (u , 42,....
ikv, N ... Vn) o ut =< u, tv, U2 +V2, ... Untuny Multiplication of scalars
Say CER U = <U,,.... Un> lcul = lalul
=<C",,CU2,... (Un> Direction of cus
same as a if (>0
 opposite of u if cco Multiplying by o we obtain the zero vector
Sly Soup
Soy Ab = OR Translate A onto o Subtract ai from ith coordinate
A (a,....an beroes Lo... o B (b,, .bo) becsues (b, a, b 92,.... bnan)
& Trus B becomes point R. o so
R(x, X2, ..... da)
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Commutativity of addition
ū+ 0 +ū
01.
+
= <u, + V,.... Un + Un >=<V,+ U
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tudy of
....
ntlln) = vt
15
30 t
ol 3
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T cos 45° + Ty sin 30 = mg end T sin 45° T Cos 30 
I = T2 13 x 2
Subtraction 03 = ( ų)
ūūū
10 lbs
sudy Soup
mg
Stay Sot
sl method 2
tudy Se
Consider
X t
o &
ů =
X
lã 1 = 1
ů is called a unit vector & is also called as the direction vector of t.

I (1 + NB) = 2mg 1 T2 = 2mg = mg (131)
(I+N3) T = 2mg 3
(IANA) 2
= mg (313) a telling a w2 ]
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CY
CYSOS
= (coso, sino) For n dimensions , say di is angle blw
a =(0, 107 a = x <cos I
Sin II)
o
X &
Ko
OU
ith
caces
U Study SOL
then
u = <Cosa , casa,.... Cosan)
6 = ß < Cos 3T , sin 37 )
= RS, 2) låt b + c = 0 & so
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& Soon
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LECTURE 3
U
wa
Stud Sap
u &u in a plame are parallel (ŭ lla) if one is a scalar multiple of the other.
ie. u = ce or w = du
where UIERE A zero vector is parallel to every vector
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f two vectors M & W are orthogonal to each other, their dot product = 0.
Is dost product actually 2 vectors I was multiplied as one is with others
component along that vector ?
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observation: Ju & W
are not parallel tur sezo
then t=s=0
othogonality
(1 ū to (2) lu ru 12 = 1 u 12 + 1012 (3) l ū  v 1 2 = 1 urt 1w 12
Proof  Let's say
tao tu = – su
u= us
o which means a llo which is a contradiction 
Whim
s=t=o
Stuc sou
To prove (1 (2)
. (ūtu). (ut) = lutul2
= u ū rū. J + v. u tu i te
= \u12+ 1012 3+2 u. V
the = lūl+ lūl?
ūū = 0 Til since ul. J=0
ulo Hence Proved.
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Dot product (also called scalar product o
 u ={u,,.... Un> wcv.... Dn> ER" u v = uU + U2 Uz t ...... + Unon ER Properties
1) Commutative : u. W = v. u 2) Linearity (CM). V = c(u. B)
u. (w+w) = uwt u w 3) ū ū= lūl?
u v = lul Tu I coso
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to find a point on a line nearest to a prescrit
PQ = Qp = QPS
brescritted bsent
Comb ab =
broja b]
=
Q Pot PP?
(P.pl
b
points opposite to a
Po is nearest et
+ is o points along a
 if Let us say ex is Kth canonical vector in IR"
x = <0,..... 0,1,0,....o)
 Kto position lexl= 1 Comp u= u er = u. e = Ux
ek je di. Find the point on y=2x nearest to the point (3 ) AI 3i
say a ↑ + 2 } a is in direction of line y = 2x
opo = tu ter Port Lino L Pop=op  opo
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Stub

to
2x
a. (ota) =0 a. t la12=0
t = ao
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1912 ector
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Po <b a ū). Sprejection of 7 on a
= projb Scalar projection of a onto a Also called the component of bo along a
Comp ba to a
lal
Benedione ab
Ta12 = (1+2 ). ( 37) di + 2
5x85 = 3 (1+y)
5 = 3/1 + b/g j
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Corellery of BCS
ane R
Kunyakovsky Cauchy  Schwarty (BCS) Inequality u We TR then
luuis lüliti
lu. W 1 = lu l l w l ifs ulla
É aiť Sna;?
X
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Equality holds if a, = 9, ..... an
(a, b, t...ta, bn ? < (ait ... + (6 +...tbr)
Equality occurs when b. = b2 =... =bno
or a ell such that a = chic & any k. (ū. %) = lūI TUT
when a 11 3 2
Either u =0
or u = cu
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1x,y,t.....tknyals Sxit.... + x 2 Jy, ... tyre Equality Reeds only iy x;= 1; = 0
or Xi = cui Consider a function Ø (t) = lut two 12 TER  $ R [o, ...)
0 (t) = 1 + tw 12 = (u't tw). (u + tw)
  1012+ tur ttu. W + t21
V = lu12+10 1 + 2 + 2 tu. w Considering & W to
DSO :. luullullol Hence broved VL
I will o then 1 u. W l = lullos
tom D=0 then 0 (t)=0 VRT. :. (utto) is a zero vector
: u llo
fa ........ an ER T I then lat....tan) 3
(a 2+ .... tan.
Studou
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