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UIUC / Mathematics / MATH 241 / Dot product is also called?

# Dot product is also called? Description

##### Description: This is a mixture of notes I made in class , made at home by reading the book and also a few of my own tricks as to how to solve certain problems. Even though it is an honors class, the notes are simple enough to help non honor students for their math 241 as well.
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studySour

## Dot product is also called? Functions of l variable

f(t)

input :

t

output : Jut)

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vector functies

Single input - multiple outputs Multivariable functions

multiple inputs - multiple

outputs

Tools

Multi - dimentional space or the n- dim. Space (n-space) R.

## Why is trn important? 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' - 3-d space (Geometacale, set of ordered toi ples of seal nos R” – set of all ordered n - tubles oleval nos

## What is tr3? 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+.....tan-bn)?

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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 + (xn-an) 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,, ..... bn-an) 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

udy sou

I

@ Study

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 co-ordinates =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

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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,.... bn-an)

& Trus B becomes point R. o so

R(x, X2, ..... da)

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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 0-3 = -( -ų)

ūūū

10 lbs

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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 (13-1)

(I+N3) T = 2mg 3

(IANA) 2

= mg (3-13) 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

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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

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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> w-cv.... 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. (o-ta) =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.

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