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So previously we talked about the requirement
of electromagnetic theory We said that there
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are certain physical phenomenon which electromagnetic
theory will explain and circuit theory will
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be inadequate for these purposes So one can
think of electromagnetic theory as some sort
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of a super set of circuits okay and electromagnetic
theory is also indispensable if you are going
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to study microwaves radar antennas fibre optics
optical communications and so on
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The primary goal of all these different areas
is to somehow harness electromagnetic energy
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and put that one into better use So electromagnetics
in one sense deals with generation propagation
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and radiation of electromagnetic energy as
well as light We will see later that light
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and electromagnetic waves are one and the
same and this union of electromagnetic theory
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and light was achieved by Maxwells equations
by Maxwell
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.
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We also studied Coulombs law which gives us
the force exerted by one charged particle
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and onto other charged particle and we have
seen the forum of this force it requires two
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parameters to specify; one is its magnitude
which is the amount of the force that is exerted
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by the particle on another particle as well
as we have to specify the direction along
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which this force acts So force is the quantity
which is slightly different or radically different
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from charge
.
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In terms of the fact that force requires both
magnitude as well as direction to specify
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whereas charge can be specified only by giving
its value number Okay So quantity is that
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require both magnitude as well as direction
are called vectors and we denote a vector
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in our hand written note by using a capital
letter okay to denote a vector and then we
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place a small arrow to distinguish that this
quantity the capital letter A with its arrow
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is different from just a capital letter
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If you dont write an arrow what it means is
that It is a scalar It can be specified by
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the number okay whereas if you write an arrow
that becomes a vector This is the hand written
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notation sometimes a different notation is
used in which you take the capital letter
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and instead of putting an arrow you put an
over bar Sometimes you also see people putting
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an under bar but that usage is quiet rare
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In text and printed materials you would find
vectors denoted by a bold phase or a bold
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print with a capital letter So here you can
see that this f vector f which could represent
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a source or some other vector quantity is
printed in bolt We also denote unit vectors
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we will discuss what unit vectors are later
So we denote unit vectors by putting a hat
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or a caret on top of the letter These letters
could be small letters for example here x
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hat indicates the unit vector in the direction
of x coordinate
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n hat indicates a unit vector in the direction
of n or you sometimes find notation by putting
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up a bold phase small letter and then putting
a subscribed to indicate the direction of
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a component So for example aR a being the
small letter but bold phase indicates that
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this is the unit vector and this R subscribed
to this letter a indicates that this is the
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unit vector in the direction of R Similarly
you will also find sometimes ux written with
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u standing for unit vector
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Again a small case letter bold and x standing
for the direction of the vector along which
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this vector is acting Now a vector is defined
graphically also by giving 2 points one is
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the tail or the origin of the vector and the
other is the end point or the final point
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of the vector or the arrow head of the vector
So this for example is a vector You have to
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distinguish between a vector which is directed
in this way
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Now if I draw an another vector that would
be pointing in a different direction these
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2 vectors will be different from each other
okay Now you also can see that we can perform
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algebra with these vectors before we can go
to the algebra let me just point out that
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in the graphical representation of a vector
the length of the line stands for the magnitude
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of the vector okay So if I have a shorter
line with an arrowthat would indicate a smaller
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magnitude vector
.
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This would indicate a larger magnitude vector
in relation to the smaller one okay Now we
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want to consider addition subtraction and
other operations on vectors for now let us
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focus only on addition and subtraction We
will come to different operations with vectors
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later so we can add 2 vectors something that
is familiar to you from your undergraduate
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studies so you have 2 vectors and how do we
add them
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So this is the recipe for adding okay I will
not go into the justification of how you obtain
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this one I hope that you will be reading up
on the reference book or you can recall from
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your earlier studies You start with the vector;
2 vectors that need to be added So you have
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a vector A which is pointing horizontally
in this way and have the certain magnitude
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okay You also have a vector B which is pointing
in a direction that is given by this blue
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line okay
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It also have the certain magnitude which is
given by the length of this line Now what
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is the resultant Or what is the sum of these
2 vectors Now before we go to the sum you
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have to understand that how did I get this
vector B okay Vectors when you are to perform
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the sum or the difference of these vectors
have to be defined from a common origin point
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okay A common origin point for the 2 vectors
A and B is for example this tail Okay
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So tail at this point will be the common point
for A as well as B Sowhich means that the
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blue line vector B should actually be defined
parallelly over here okay This is the 2 vectors
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which are acting and the vector C the red
coloured arrow is the resultant of A and B
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but to calculate the resultant or to calculate
the sum of the 2 vectors what I have done
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is; to take these vector B and translate it
parallelly to get this vector B which is the
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same vector
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But this vector has its tail at the end point
of A So this is the first recipe you can actually
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translate a vector parallel in a plane that
you want The 2 vectors are mathematically
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equivalent however physically they may not
be equivalent For example I have this object
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here okay now I apply a force along this direction
so now you can see that this is the starting
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point of the force and this is the end point
of the force let us assume that
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This vector is making contact with this object
at this point So this is the vector that is
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acting at this particular point Now mathematically
you can represent this vector by a line with
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this as the origin and this as the end point
and that could be a vector that would look
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like this Now you can translate it parallelly
mathematically a vector that is that I am
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showing here and a vector here both are equivalent
however physically these 2 are not at all
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equivalent
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This vector is not all acting on these object
or you could translate it down okay in the
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same plane again the 2 vectors are mathematically
equivalent but the 2 vectors are not physically
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equivalent okayThe reason why I wanted to
specify this is because we sometimes get carried
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away by mathematical operations okay you can
actually translate a vector parallelly along
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a plane
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But then you have to always constantly remember
that these vectors are not physically equivalent
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however mathematically you can translate them
you can add you can subtract and the resultant
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vector will always be correct okay So with
that aspect in mind you have a vector B which
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was originally defined from the common point
of the 2 vectors A and B having been parallelly
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translated over here okay
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So we have parallelly translated the vector
B Now I want to find the resultant of A and
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B this is the second step of the recipe The
first step is to parallelly translate a vector
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the second step is to now add the 2 vectors
how do we add the 2 vectors The sum of the
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2 vectors which in this case sum of A and
B is C and this is given by a vector which
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has its origin at the tail of A and at the
head of B So you can see that the red arrow
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goes from the tail of A and all the way up
to the head of A okay
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I have said that you can translate a vector
parallelly and they will be mathematically
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equivalent and you can see that over here
as well So I can take the vector A I can leave
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the vector B as it is Now I can take the vector
A and translate parallelly to get a vector
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which is this dashed vector this is also labelled
A because mathematically this vector and this
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vector are equivalent so I have translated
this vector parallelly such that the tail
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of A is at the head of B
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Now I have one tail here and one head here
but you can actually now joint the tail of
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B to tail of A to get the resultant vector
A+B which is the sum of the 2 vectors A and
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B but please note carefully that this vector
is the same as this vector because you can
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see that these 2 vectors are parallel to each
other okay This is the law of triangular addition
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for 2 vectors and this is something that you
will have to practice a little bit to get
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familiar with this sum or addition of the
2 vectors
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Because we will be looking at not just 2 vectors
we will be looking to add more than 2 vectors
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Now how do I add more than 2 vectors You do
the recipe pairwise that is; if I have 4 vectors
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to begin with call ABC and D First I find
the sum of A and B then I find the resultant
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of the sum with C and then I will add the
resultant of these 3 vectors to the next vector
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D and I can keep doing this one until I get
tired right
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So lets do that one for 3 vectors and you
can actually see that how the process works
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so you have a black line which is representing
a vector A here is its tail here is its end
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or the head of the vector A To this vector
I want to add the vector B okay B is this
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blue line vector which is now added so as
to get the resultant A+B so this A+B is the
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red vector over here okay So I hope this is
all right you have a vector A you place the
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tail of the vector B to the head of vector
A
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Then you draw a line from the tail of A to
the head of B so as to get the resultant A+B
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okay Once you have the resultant vector A+B
now I want to add to this A+B vector a different
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vector called C okay So now C is the orange
vector over here I am going to add A+B to
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C how do I do that I have one vector A+B if
C is not originally at this place you can
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translate C somehow if c was a vector that
was defined elsewhere on the plane you parallelly
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translate the vector until the C tail reaches
the head of A+B
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Now you can forget about A and B and you can
simply concentrate on the 2 vectors A+B and
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C I have A+B vector and C vector you now draw
a new line which goes from the tail of A+B
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to the head of C okay So this way you can
add 3 vectors and this green line or the green
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vector is the resultant or the sum of all
the 3 vectors okay So I hope this process
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is convincing you can show that vector addition
is commutative as well as associative by looking
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at A+B or B+A
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So you can show that A+B will also lead to
the same vector C or B+A see the blue line
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and the dashed black line will also lead to
the same vector C which is just a parallel
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translated version of this solid red line
okay So vector addition is commutative to
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show that vector addition is associative that
is whether you start with A+B+C if this is
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a sum of the vectors that you want You can
first find the resultant of A and B and then
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find the resultant of A B; that is A+B and
C vector
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You can do that one by going first to A and
B to get A+B to that you add C to get A+B+C
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Alternatively you can start with B+C here
that would give you a vector from the tail
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of B to the head of C okay and from here you
can then add a vector A you will again get
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back to the same vector you can try this an
exercise to convince yourselves that vector
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addition is commutative as well as associative
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So far we have talked about addition of vectors
we will also need subtraction of vectors we
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also need to subtract one vector from another
vector so how do we do This operation how
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do I subtract B from A remember that subtraction
of vector is equivalent to addition to the
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vector A the negative of B that is A-B is
equivalent of A+(-B) okay To get to that one
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you need to know what is –B –B is a vector
which has the same magnitude as vector B
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But it will be pointing in the direction opposite
to that it will be pointing exactly opposite
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to the direction of B and has the same magnitude
remember magnitude is the positive quantity
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okay So vector B which is defined from this
point along this direction has the same length
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as the vector –B except that vector B is
pointing in this direction whereas vector
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–B is pointing exactly in the opposite direction
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Now to add 2 vectors A and –B is very simple
you take a vector A you place the vector –B
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the tail of vector –B at the head of vector
A and then you draw another line from tail
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of vector A to head of vector –B so you
get A – B which is you can call as a different
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vector D I hope that you are convinced now
that the vectors can be added and subtracted
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now if you are asking or thinking about how
do I multiply the 2 vectors
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We will have to wait for that for sometime
okay We dont really need the multiplication
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at this point we will wait for sometime before
we tackle the subject of multiplication okay
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So far we have talked about vectors in a very
very general way okay We drew some lines then
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we added lines to another line and we found
the resultant line so we did all these things
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however this was all the graphical method
of doing vector analysis or vector algebra
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okay
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You might keep doing graphical methods of
vector analysis but then you will soon tired
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yourself out if the number of vectors involved
is more than 3 okay For example in certain
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problems you will be seeing that there will
be charges which are say 10 or 20 placed around
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in a plane and if you want to add all these
resultant force because of all the individual
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charges so you will be adding 20 such forces
may be all of different magnitude and different
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direction
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So it will become tiresome if you want carry
out the graphical analysis though graphical
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analysis helps you in visualising how the
vector would look Okay So we need an algebraic
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method of handling this this algebraic method
of handling vectors can be accomplished when
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we introduce a coordinate system okay So what
coordinate systems do is that they help you
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specify the locations in space by numbers
as well as describe a method in which a vector
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can be equivalently described by a set of
numbers
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.
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Now this is the great advantage a vector which
could be oriented in a random direction in
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the space is now being described by a set
of numbers and if you also find out how to
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add or subtract vectors with just these numbers
then we can not only do this as the algebraic
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exercise we can also put in onto a computer
okay input this to a computer so that the
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entire thing can be computationally performed
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So you will actually save a lot of time if
you are able to associate numbers to vectors
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and coordinate systems are the ones which
give you give you those numbers Now before
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we proceed further I want to clarify one very
important aspect Coordinate systems are independent
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of vectors in the sense that if I change the
coordinate system the corresponding vector
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the vector will not change however the numbers
that we associate with the vectors will change
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okay
.
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This soon become apparent to you we will go
with some examples as you can see but please
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keep in mind that coordinate systems and vector
themselves are independent The vector does
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not depend on the coordinate systems for it
existence however the numbers we attached
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to the vectors are dependent on the particular
coordinate system that I have chosen or you
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have chosen okay But normally we choose certain
standard coordinate systems and work with
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that so that the minimal there is a minimal
amount of confusion