1Vector and Its Operations Lesson 1

8/6/2019 1Vector and Its Operations Lesson 1

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Vectors and the Geometry

of Space

Chapter 1


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Vector and its Operations

Lesson 1

Objective:

At the end of the lesson you should be able to:

1. Define vector.

2. Find a vector , its magnitude and its direction.

3. Use the fundamental operations of vectors.


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Vectors

Definition:

Vectoris a quantity with magnitude and direction.

Example: displacement, velocity , weight, force

How to find a vector?

1. Given two points in the xy-plane

2. Given two points in the xyz-plane

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"!"! bayyxxv

yxPyxP

,)(),(

),(,),(

1212

222111

"!"! cbazzyyxxv

zyxzyx

,,)(,)(),(

),,(,),,(

121212

22221111


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The components of the vector in the two dimensional

space are a and b. While in the three dimensional space,the components are a, b, and c.

Addition of Vectors (Graphical Method)

1. Triangle Law 2. Parallelogram Law

The sum is the resultant vector.

u+vv

u

u+v

v

u

v

u

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Addition/Subtraction of Vectors (Analytical Method)

Given two vectors a and b where a =< > and b= < >

a+b= < > and b-a=< >

Likewise ifa = < > and b = < >

a+b=< > and b-a= < >

Magnitude of a Vector

The magnitude or length is indicated as

"!!

"!!

321

2

3

2

2

2

1

21

2

2

2

1

,,,)()()(

,,)()(

aaavaaav

aav

aav

For two and three dimensional vectors respectively.

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21

,aa21

,bb

2211 , baba 2211 , abab

332211,, bababa

321 ,, aaa 321 ,, bbb

332211 ,, ababab


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Multiplication by a Scalar(c)

A vector can be multiplied by a scalar quantity c . If vectora=< > then ca = < > or if vector

a= < > then ca = < >.

Properties of Vectors

1. a+b= b +a 2. a+(b+c)= (a+b)+c

3. a+0= a 4. a+(-a) = 0

5. 1a=a 6. c(a+b)= ca + cb

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21, aa 21,caca

321 ,, aaa 321 ,, cacaca


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The Forms of a Vector

A vector in two dimensional takes the i andj form .For

example, a = 2i 4j, in component form this can be

written as, a= .

A vector in three dimensional space takes the i, j, and k

form. For example, a =-3i+ 6j- k , in component form

this can be written as,a = < -3, 6, -1>.

Any vector can be represented by a small bold letter or it

could be represented by with an arrow above the

letter.a

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Examples

a) C( 3,-5) and D( -6, 2)

b) b) C ( 1, 3, 4) and D ( 5 , -1, 0)

find the components of the vector CD and itsmagnitude.

a) v= = = -9i + 7j

b) v= < (5-1), (-1-3),(0-4)>= < 4, -4, -4> =4i- 4j- 4k

1307)9(

22!!

v

1. Given are two points C and D

34)4()4(4222 !!v

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Examples

2. Given a= < 1, 2, 3> and b = < -1, 3, -6> find the

following:

a) 2a +b b) b3

2

Solution:

a) 2a + b = 2 + = =

"!"! 4,2,3

26,3,1

3

2

3

2bb)

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Solve 21 and 22 on page 841!!


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Unit Vector

Definition

A unit vector (u) is a vector whose length is one. For

instance, i ,j , and k are all unit vectors.

In general ifa # 0 then the unit vector that has the same

direction as a is

a

a

au !!

1

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Do problems 24 and 25 on page 841


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Example 1: Unit Vector

Find the unit vector that has the same direction as the given

vector:

a) v = < 3,-6 > b) a= 5j- k

Solution

a)

45

6,36,3

6,3

1 "!"

"

!u

b)26

1,5,01,5,0

1,5,0

1 "!"

"!u

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The Dot Product

Definition

If a = and b = then the dot

product ofa and b is given by

"321 ,, aaa " 321 ,, bbb

332211

321321 ,,,,babababa

bbbaaaba!y

"y"!y

Example 2: Give the dot product of the following:

.4,1,36,3,2)

5,18,4)

"y"

"y"

b

a

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Answers:

a) 36)5(8)1(45,18,4 !!"y"

b)

212436

)4(61)3()3(24,1,36,3,2

!!

!"y"

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Solve more problems on page 848


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Angle Formed Between Two Vectors

Theorem

If is the angle between the vectors a and b, thenU

U!y cosbaba

Corollary

If is the angle between the nonzero vectors a and

b, then

U

ba

ba y!Ucos

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Direction Angles and Direction Cosines of a Vector

Definition

The direction angle of a nonzero vectora are the

angles in the interval [ 0, ] that a makes

with the positive x, y, and z axes.

KFE and,, T

z

y

xE

F

K

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Use the following formulas to find the direction cosines

"!

!K!F!U

321

321

,,

cos,cos,cos

aaaawhere

a

a

a

a

a

a

and the direction angles are

a

a

a

a

a

a312111

cos,

cos,

cos

!K!F!U

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Example 3: Angle Between Two Vectors

1. Find the angle between the two vectors a = < 1, -2, 3 >

and b = < 0, 5 , 9 >.

Solution

01

22222

8.7340.0cos

40.0

45.42

17

10614

27100cos

9503)2(1

9,5,03,2,1cos

!!U

!!

!U

"y"!

y!U

ba

ba

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Example 4: Direction Angles

2. Find the direction angles of vector b =3i + 4j + 5 k.

Solution

01

1

222

1

65424.0cos

50

3cos

543

3cos

5,4,3

!!E

!

!E

"!

b

What are the values of ?KF and

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YourAnswers?

011

01

45707.0cos

07.

7

5cos

5.55

07.7

4cos

!!!K

!!F

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Orthogonal Vector

Definition

Two vectors a and b are orthogonal if .

Recall that and if the angle is

90 degrees its cosine is 0 hence the dot product between

vectors a and b is 0.

Look at some examples!!

0!yba

U!y cosbaba

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Example 5: Orthogonal and Parallel Vectors

Find out whether the two vectors are orthogonal or parallel.

1. a = and b =

2. u = < 1,-1,2 > and v =

Solution

1. The two vectors are not parallel ( parallel only if they

have the same direction i.e., the components of the

vectors are the same or b=

ka ).

The two vectors are orthogonal.

05831,4,35,2,1 !!"y"!yba

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2. 52121,1,22,1,1 !"!y"!y vu

The two vectors are not orthogonal neither they are

parallel.

Do some more problems on page 849 nos.

23 a, d and 24 a , c.

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Remember!

The dot product is a scalar.


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Cross Product

Definition

Ifa = and b= then the cross

product ofa and b is the vector

"321 ,, aaa "

321 ,, bbb

"!v

!v

!"v"!v

122113312332

122113312332

321

321

321321

,,

)()()(

,,,,

bababababababa

babakbabajbabaiba

bbb

aaa

kji

bbbaaaba

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Theorem

If is the angle between a and b then .U U!vsi

nbaba

The cross product is the vector that is perpendicular to

both a and b.

Corollary

Two nonzero vectors are parallel if and only if

Take note that if sine of these angles are 0.

Also the magnitude of the cross product is equal to the

area of the parallelogram determined by a and b.

0!vba

T!U!U or0

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a

b

U

Usinb

Using Figure 1, A = baba v!U)sin(

Figure 1

For a parallelepiped (rectangular solid) volume is

determined by using scalar triple product i.e.,

If V = 0, then three vectors are co-planar.

)( cbaV vy!

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Example 6: Cross Product

1. Find the cross product given a = and

b = .Verify whether it is orthogonal to both a and b.

Solution

kjikjiba

kji

ba

32)03()01()02(

130

0211,3,00,2,1

!!v

"!v"!v

Then to verify whether orthogonal to both a and b,

00

330022

1,3,03,1,20,2,13,1,2

0)()(

!

!

"y"!"y"

!yv!yv bbaaba

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2. Find the volume using the scalar triple product given

a=

i+

j

k, b=

i-j+

kandc =

-i+

j+

k.

Solution

4022)(

11

111

11

111

11

111

111

111

111

)(

!!vy

!

!vy

cba

cba

The volume of the parallelepiped is 4 cu. units.

More problems can be found on page 857. Practice

by solving some of them!

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Torque

Torque is the turning effect when a force is acting on a

rigid body at a point given by a position vectorr.It is defined

to be the cross product of the position and force vectors

i.e.,

U!v!X

v!X

sinFrFr

Fr

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Example 7: Torque

A bicycle pedal is pushed by a foot with a 60 N force asshown. The shaft of the pedal is 18 cm long. Find the

magnitude of the torque about P.

60 N

P

P

JoulesNm

Nm

64.1064.1080sin)60(18.0

)1070sin(6018.0

0

00

!!!X

!X

010

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Solve36 and 37 on page 857!

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1Vector and Its Operations Lesson 1

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