vectors lesson 1 leaving certificate ordinary level option st. joseph’s cbs maths department...

VectorsVectorsLesson 1Lesson 1

Leaving Certificate Ordinary LevelOption

St. Joseph’s CBS Maths Department Fairview Dublin 3

M Timmons

Wednesday 19 April 2023

Each lesson may take a number of classes.

The movement from The movement from aa to to b.b.

Definition of a Vector: A vector has both length and direction.

ab


Same length same direction.Same length same direction.

Equal Vectors

ab

cd

cdab Here


The vectors are equal even The vectors are equal even though they are in different though they are in different locations.locations.

In the parallelogram abcd

bcaddcab and Here


Modulus also called absolute Modulus also called absolute value.value.

Modulus of a vector: is the length of the vector.

ab asWritten

cm6 Here

ab


To get rid of the minus just switch To get rid of the minus just switch the lettersthe letters

Negative Vector: is a vector parallel and equal in length, but going in the opposite direction.

abbaab

ba

isThat . vector the

of negative thebe tosaid is vector The

cd dc- generalIn


Negative scalars the arrow goes in Negative scalars the arrow goes in the opposite directionthe opposite direction

Multiplication of a vector by a scalar.(Number)

a

a2

a5.1

a

The 2 ,-1.5 etc are called scalars


The result is a vector that goes The result is a vector that goes from the beginning of the first from the beginning of the first vector to the end of the second vector to the end of the second vectorvector

Adding Vectors

Triangular Law: Put the start of the second vector on the end of the first vector.

ba


The Triangular Method and the The Triangular Method and the Parallelogram Method produce Parallelogram Method produce the same vector.the same vector.

Adding Vectors (Continued)

The Parallelogram Law: Both vectors must have the same starting point. Complete the parallelogram. The diagonal from the starting point is the sum of the two vectors.

a

b

ba


In parts (i) to (iv) the Law is used. In In parts (i) to (iv) the Law is used. In part (v) the Law is used. part (v) the Law is used.

Example 1

adba )i(

abda )ii(

bmab )iii(

mcam )iv(

cdcb )v(

bd

db

am

ac

ca




St. Joseph’s CBS Maths DepartmentFairview Dublin 3

Mr M Timmons


This is a very important result and This is a very important result and is used frequently when dealing is used frequently when dealing with vector problemswith vector problems

Single Letter Vectors

aoa

bob

coc

Important Rule

abab The second vector minus the first vector

aao


Notice the Rule Notice the Rule

Example 2

b )i(

ac )ii(

om )iii(

caaboa

Rule

ac

caob2

1

2

1

2

1

Express in terms of and

c

a

am iv)(

acac2

1

2

1

2

1

ababWednesday 19 April 2023

The diagram above shows two The diagram above shows two perpendicular axes, just like the perpendicular axes, just like the xx and and yy axes. axes.

The Perpendicular vectors

ji and

The vector

ji3


yourself 3- and 4Graph

jiji

You can be asked to show the You can be asked to show the result on a diagramresult on a diagram

Operations with the Vectors

ji and

(i) Adding and Subtracting: you add (or subtract) the parts

and parts separately.

i

j

jibjia 3 and 43

4

jiba 72

jiba


This is the very important result that I talked This is the very important result that I talked about earlier & is used frequently when about earlier & is used frequently when dealing with vector problemsdealing with vector problems

(ii) Multiplication by a Scalar

jibjia 2 and 23

jijia 69 2333Then

jijib 3623 3- and

(iii) The Vector

abab

jibjia 2 and 23

jijijiabab 35232


Here we are solving two simultaneous Here we are solving two simultaneous equations. equations.

Equality of Vectors3 and 1Then 3 If

bajijbia

bajijibjia and Find 127 2 If

Ex 3

7ba

122 ba

71 ba122 ba

7 ba122 ba5 a

7ba

257

75

bb

b


jijbibjaia 127 2

ia

ib

i7

ja2

jb

j12

Ex 2




M Timmons


The Modulus is the distance from The Modulus is the distance from the origin to the point (the origin to the point (a,ba,b))

The Modulus of a vector in terms of


ji and

22 as written Also yxjyix

22 vector theof modulus The yxjyix

:find 32 and 86 If

jibjia

Ex 1

a (i)

b (ii)

ba (iii)

1010086 22

1332 22

895858 22

ji

To get the related perpendicular vector To get the related perpendicular vector we swop the and coefficients and we swop the and coefficients and change the sign of the change the sign of the

The related Perpendicular vector

r

then If

jxiyrjyixr


i

j

i

Ex 2

:find 3 and 43 If

jibjia

a (i)

b (ii)

ba (iii)

ji 34

ji3

1741-4 22

ji




M Timmons


This last formula gives us the This last formula gives us the angle between any two vectors angle between any two vectors

Dot Product (There are two definitions)


θcos. (2)

yxyx

bdacyxjdicyjbiax

. Then and If (1)

yx

yx

yxyx

. θcos

θcos. (2) From

Angle Between 2 vectors

Don’t confuse with Don’t confuse with

Perpendicular Vectors


yx .

xy

By Definition (ii) θcos.

yxyx

0090cos.

then,90 is and vectorsobetween tw angle theIf

0

o

yxyxyx

yx

If the dot product of two vectors is zero, this implies the two vectors are perpendicular to each other.

0..other each to are and Because

ijjiji

When the dot product is zero the When the dot product is zero the vectors are perpendicularvectors are perpendicular

Ex 1


Evaluate the following:

jiji 62.43 (i)

bdacyxjdicyjbiax

. Then and If (1)By

30642362.43

jiji

jiji . (ii)

01111. jiji

Here we use the inverse cos button on your Here we use the inverse cos button on your calculator to get the angle. calculator to get the angle. Shift cos( 4÷5)=Shift cos( 4÷5)=

Ex 2


jibjia 24 and 2 Given

a (i)

jiji 24.2 82241

Find:

b (ii)

ba . (iii)

ba and between angle The (iv)

521 22 2024 22

5

4

100

8

205

8. θcos

ba

ba θ5

4cos

5

4θcos 1 087.36θ

Vectors is your option for Paper 2 Vectors is your option for Paper 2 of the Leaving Certificate. You do of the Leaving Certificate. You do need to know this Material.need to know this Material.

End of Topic


vectors lesson 1 leaving certificate ordinary level option st. joseph’s cbs maths department...

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