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Vectors in Linear Algebra: Lecture Notes and Exercises, Lecture notes of Engineering Mathematics

Lecture notes and exercises on vectors in linear algebra. It covers fundamental concepts such as vector components, dot product, cosine theorem, and cross product. Illustrative examples and explanations to enhance understanding. It is suitable for students studying linear algebra at the university level.

Typology: Lecture notes

2023/2024

Uploaded on 12/25/2024

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WEEK 2LECTURE 2
Vectors in 1123 VLVe Vz Vz Yi Vz jtVz K
ice
componentsof
vector v
2
Panik b
tap
Ji
hs Vz j
Dotproduct another operation betweentwo vectors
but this time the result is aScaf
Given hi CUn Uz Uz Up it 42 jt43 k
vLVp Vz Vz Va it Vz j1Vz k
utf uvta zVz't 43 3
Properties aaKvu
ca uului 14 magnitude
Indeed ULun 4443
2
UUUp Up1Uz 92 43.43 4,2 42745 1UI
pf3
pf4
pf5

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WEEK 2 LECTURE 2

Vectors (^) in

1123 V L

Ve (^) Vz (^) Vz Yi Vz j tVz K ice componentsof vector v 2 Panik (^) b tap J i hs Vz (^) j Dotproduct another (^) operation between^ two^ vectors but this time^ the^ result^

is a

Scaf Given hi^ C^ Un Uz^ Uz Up it^42 j t (^43) k v L^ Vp Vz Vz Va^

it

Vz j 1 Vz k u tf^ u v^ ta (^) z (^) Vz't 43 3 Properties a^

a K^ v^ u

ca u^ u^ lui (^14) magnitude Indeed U (^) Lun 4443 2

U U^

Up (^) Up 1 Uz^92 43.43 4,2^42745 1 UI

3 u Vfw^ u^ v^ tu^

w t

vectors

The cosine^ theorem 2 Given two^ vectors^ uandu we have

a v^ 14.1 1

cost tangle between u and Proof B 11342 1 ABFTIACT n

UY

ya u 21m31^ Hel^ cost 1T Thom 5 generalized^ Pytagorean A W^ C

AB u^ AE^

V BI r u

12 142 142 214 NIMT

V a^ V^ a^ u^ u^

v.v 214 Walcott

v u^ u^ v^ u^ u^21414 cost Zu V^ 21h^ lv

cnTu.oV1hl.lVlcos

O u^ V^ C^1 Y 12kt^ 5kg For (^) example 4 0

will work 3

Answer (^) v Co^5

Cross

product an (^) operation between^ vectors^ giving us another vector^ as^ a^ result u (^) Car (^) 92,937 9 i^ ta^ j

ta z k

be bz^ bz ba.itbijtb.sk 11 11 DEI (^) uxv Ya ta Groan (^) product be bzb faabzazlfi cqbz^ ghjtcqbza.TK Properties (^) e axv Vau (^2) Uav is (^) perpendicular to both^ and (^) VL Indeed we^ will^ check^ that xx^

u o

UN V^ O

u xv^ azbz azbdii cqbz azbdj cqbz

az.ba K

U

9 I^ t^92 j t 93 K

UN u a (^) azbz azbd adaebz^ azbdtazca.bz^ AID 91913 919382 9,913^92938

101 i Remake Ju The tight hand Mule (^) applies I

v u^ UN

ft

luxuf lul.lv isinFangeeueatauedey a consequence (^) of anime theorem Example let u^

C

2, 17 2 it^ 3J

th k

L 1 I o^1 i^ l^ j

to K

w 7,3 z^7 it 3 j

t 2 K

compute u YY