Scalars and VectorsOutline: - Scalars , vectors (definitions)- Representing vectors- Vectoral Operations

addition, subtraction, multiplication (with a scalar)

- Coordinate systems* Cartesian coordinates* Polar coordinates

- Multiplication of vectors* Scaler multiplication (Dot product)* Vectoral multiplication (cross product)

- Physical parameters are split into two big classes: Scalars and Vectors

Displacement, velocity, acceleration

- A scalar quantity is described by a single number (with an appropriate unit). This number is defined as Magnitude of the scalar.

- A vector quantity has both a magnitude and a direction.mass, time, energy, temperature

Representing a vector

Addition

- We show a vector as and denotes its magnitude.- Draw a vector as a line with an arrow head at its tip. The direction of the arrow points the direction of the vector.

- The length of the vector represents its scalar magnitude.

arrow = direction If a quantity has also a certaindirection, then our results should also indicate its direction. Thus we need to use vectors in ourtheoretical calculations and experimental measurements.

length = magnitude

Now the question is how we can use the vectors in our calculations.

Vectoral Operations

Before discussing the details, let us first consider vector addition diagrammatically. It helps to present resultant vectors as a regular vector i.e with a line and arrow head at its tip.

There are two graphical methods which we can use in vector addition.

qA

orB

d

9A

cgao

Bd

7AaD Toa ED

9

B cr

ParallelogramIn order to add two vectors, they have to be in contact. However, the vectors are not given in contact. Thus, we need to move them. We can move a vector from one place to another without changing its magnitude and its direction.

To use parallelogram method, we need to move vectors such that their tails will be in contact.

You can change the order of vectors in the addition. The result will be the same.

What if we need to add three vectors?

You can add only two vectors at each step, and you need to repeat twice to add three vectors with parallelogram.

E I iBtDA 9 A g F tip9

dB

Br

E BtAtBA 9rItBtD

Ba ofD

E

Head-to- TailIf we follow the method called head-to-tail, we can add as many vectors as we want in the vector addition. In this method, a vector is involved in the vector addition by attaching its tail to the head of another vector.

The order of the vector does not change the result. Thus you can add these vectors in any order.

F I Ba a9 A

A

Fe

B dBd

8 5 If a n n E E

orS F

G 6

B E B FI r AI i3

inF

E E

Vector Subtraction

The order of the vectors is important in vector subtraction.

and have thesame magnitude (same length), but oppositedirections.

Two vectors are said to be equivalent if they have the same magnitude and same direction. A minus sign in front of a vector reverses its direction but keeps its magnitude the same.

I

IFI B a At IB t laAIlal.LA

B At

SFI pf Fg

lat.INT

B aA lal.A lal A

A7

Be

IBT lallAT

Multiplication

a) Multiplying a vector with a scalar

If we multiply a vector with a scalar "a", it changes the magnitude of the vector.

What about the direction?

* If a is positive (a > 0), then and will be in the samedirection. Hence, we can show them as

* If a is negative (a < 0, a = -|a|)

reversing the direction

I

Agg IIAT Vaftaf

IAx Gso

Ax Ay lATSino

ya i jlil 151 1far xi 1JA A it Aaj

We will need to multiply vectors with another vector in our calculations. Let us first introduce another tool that will be quite helpful in our calculations.

- Cartesian Coordinate System

Coordinate System

in 2-dimensions:

How are we going to use the coordinate system in vector operations?

unit vectors

n't Bz R EtBBy z i pTzRxitRyjRaft Fa iAa t R AatBx

I l t

AyyeBy x Ry _AftBy

Rx

Ln E j k

qlil ljl.lk

1fIhszitjLkI A AxieAyJtAzK

ez B _BxitByjtBzk

Magnitude of the vectors I _AIBHII _VATtAJAIR R.it

2yjtRzI1BlVB BytBzRx Ax BxRy _Agt By

IRT_VREtRytRz Rz AztBzINITIATHBY

Easy to generalize to 3- dimensions

unit vectors

I

orAE Be III III Caso

lorB

C tI.B Fibc

i I _I I i O I i 0

i j O j.j 1 I j 0

i I D j k 0 I I L

At _A ith yjAIKBBxi Byj.BZA.B LAei Ayj Azkl.LB.i Byj Bz.li

AxBx i E AXByli.jc A.Bzci.liAybxlj.il AYBylj.jli AYBzcj.laAzBXLI.il ATByck.jl AZBr LI.la

ATB AxBxtAyBytAzBz

Scalar Multiplication (Dot Product)

The result of the scalar multiplication is a scalar quantity!

vectors

scalar

Applying scalar multiplication to the unit vectors

Let us now calculate the scalar multiplication of two vectors in terms of the unit vectors.

C I I C III III Cosac AxBxtAyBytAzBz

CosABxtAyBytAz_III 1151

I _Fx RT let _III 151 Sino

C

A

Bz

E AIB BxA

D E

Angle between two vectors

Vectoral Multiplication (Cross product)

However, the result of the vectoral multiplication is another vector. Thus we need to determine the direction of

Right-hand Rule

1- Raise your right hand2- Keep your thumb close3- Put other four fryers in the direction of the first vector in the vector multiplication (here it is )4- Adjust your palm such that your palm points the direction of the second vector (here it is )5- Finally, open up your thumb, it will show you the direction of the resultant vector (here it is ).

If you switch the vectors, the result will be in the opposite direction.

Exi O jxi I Ixi jixj k jxj 0 Teej iixia j jxk i Taxi _O

n

Cr CtlHe H unr a J

n CKr

Ct

At _A itAyjAZIBBxi Byj BZI

AXB CA.ci iAyjtAr Bx 3zIAXBxlixii AXBylixjl AXBzli.liAYBxljxil AYBycjxjl AYBzlj.liAZBxtk.ci tAzBynkxgltAzBzCkxk

AxBylk tAxBzC jAyBx C E Ay BeliA Bx J Arby I

The vectoral multiplication of unit vectors:

Cyclic property:If you follow the black circle, the result will be in the positive direction.

If you follow the red circle, the result will be negative.

I y l

tIxB AgBz AzBy I c AzBx A Bz jAxBy AyB I

A'xD Cai 1cg Jt CzkG AyBz ArBy Cy AtBx AxBz Ct A By AyBx

d d H

AI B Y T KI 1

AY Ay Az1 IBix By Biz1 I

n Ay Az n Ax Az n Ae Aye X J X ek XBy Bz Bx Bt Bx By

i AyBz AzBy j A Bz AzBx tk AxBy AyBx

Calculating the vector multiplication with determinants

Bi

D E

A e

Ba AT BI AT

BcAB Doo E

A AcAc

BFB to BE DPT BET DFTD

DE E

and ART 2 DFBT 2 BET

AT 113T2DBt2BTEJ

AT 2DEABT BI 2 DRT BFI and

AT DTE 1AIl 2lDTEl orIDEK 1442

Solved Examples

Let ABC be a triangle and let D be the half point of AB and E be the half point of BC. Prove that |DE| is of length which is half of |AC|.

Example: (mid-point theorem)

Solution:Let us first assign a vector to each side of the ABC triangle:

Now assigning a vector to each side of the DBE triangle:

(D is the mid-point)

(E is the mid-point)

A 3 i Sj 16kB 2E 3J 13k

F B IAT.LIrGsOrIrt5AxBxtAyBytAzBzIAT

VAxtAy4AE 35tt55tC6 8.4

IB t IBxh ByeBET VC 25 C 35 cz

4.7AeBxtAyByAzBt 37 C 2 t L S 37 161.13

27

Cos AxBxtAyBytAzB271AT 151 8.4 4.7

cos 050.03 30

Example:Given the vectors as

Find the angle between these two vectors.

Solution:

IAI 1 5 11371 4 37

ArefB Area IAI x BTI3 LARTI IBTI SiroCcl S 41 Sin 77A CArea Area I 12.03 unit

Example:Calculate the area of the triangle given in first example if

and the angle between them θ =

Solution:The area can be calculated with the vector multiplication of the sides. Thus

If the area is represented with a vector, its direction will be perpendicular to the plane of the triangle.