Component, magnitude of vectors - Vector Algebra Video Lecture - Class 12

hello friends this video on vector
algebra part mine is brought you by exam
fear calm no more fear from exam before
watching this video please make sure
that you have watched part 1 to part 8
now let's discuss the most critical
chapter a critical module or the vector
components of a vector generate vector
in this case so we have there suppose we
take three points a B and C a B and C
three points a half with the points 1 0
0 0 1 0 I or C 0 0 1 so I have three
points give me Z let's write this in the
vector form so if you see this vector
point a first
this guy is 1 0 0 so for this vector I
will say a vector in the X direction it
has full contribution why because this
whole vector is an X direction only
so 1 I the length of this vector in X
Direction is 1 to 1 I Y & Z there is no
contribution because they are this is
all perpendicular to y&z direction 0 J
camp mm 0 keycap this guy is nothing but
I cap
this guy a vector is nothing but a unit
directional unit vector in X direction
so it is listened to by I can see we
talk about B vector the point is 0 1 0
so this B is 0 1 0 if you see for B
vector in I direction there is zero
contribution in Z Direction full
contribution and K Direction again zero
contribution this guy is nothing but as
a vector this B vector is nothing but a
gyrator is also unit vector 1 length
sniffer talk about C vector this guy the
zero contribution I 0 is a and one full
in k this guys shave it this is also a
unit vector correct does it you see all
these lengths oh a OB and OC are one
unit
cut and this follows our right hand rule
XY if you rotate on here to where you
can Z these should keep on checking for
all the questions because it is critical
that we follow the right hand rule right
and these vectors are all one unit
vector in X Y & Z so we are talking
about three vectors here my Oh a vector
OB vector and possible right all these
enter Oh a Joaquin you see our unit
vectors in X Y and n direction
now if I have some position vector if I
have some position vector here let's
suppose P so this guy will have some
component in X Y in Z direction so we'll
see this so let's suppose my P vector is
x y&z points or XY and Z vector so this
P vector is denoted by our C in this
case this particular position vector can
be written as our vector where this is
nothing but X I plus YJ plus ZK where if
you see x y&z are the components of this
vector on XY and z axis because this
point is XYZ you see this guy is my
vector P now if you want to find how
much effect this guy has on let's
suppose Z X's so this guy is the effect
this guy will AHA said Isis right just
much how you see this guy from here
this guy is my Z similarly for y how
much this guy will have and YX is you
take the cause of this you get this much
right because we're talking about one
triangle right this guy is y similarly
when you talk about this vector the
effect of this vector on x-axis this guy
is nothing but the X component of the
so that is X I okay because this vector
is my this waiter it starts from here
it touches here so this guy will have
this much impact on x-axis this much on
Y and this muscle there so this guy is
nothing but X I plus y Z plus Z K
correct so this is the form of vector is
called comparing form where you break
this vector into X component Y component
and set combo right this guy is a scalar
this guy's a vector all the vectors X I
YJ and ZK then always the x y&z are
scalars they are all scalar components
and X I YJ and ZK these are all but
because this guy is vector i i've Ector
Z vector k if it all are unit vectors in
XY and z direction in that you're
multiplying the scalar so you get this
these vectors this is one vector this
one vector and this guy's correct these
are the vectors you get this is the
vectors you get X Y Z this particular
vector R vector can be decomposed into
vectors in along X Direction y and z
direction that's what we like this so
the same vector R can be represented as
X plus y is a plus ZK where x y and z
these all are I can say X I Y Z and
inside K are three different vectors you
add all these XY and z vectors you get
all negatives correct mathematically we
can prove that you have X I plus y is L
if you add you get just this guy ok I'll
show you you say X plus y Z you get a
sorted line on this dotted line you add
okay this guy is that K does if you see
this guy is that K this guy this guy oh
so that K P when P so on this you add Z
K that is X I plus y is a plus that K
you get this vector that is X plus y Z
plus second correct by by the vector
addition or we can see that you add
these two vectors X I Y Z and Z K right
is X I Y Z and Z K you get the R vector
that is that is what X I plus y z+ it
hard to get this ux i plus y z if you
add you get this vector correct this
vector and this is your again Z K vector
you add K vector here you move this the
K vector so what you get here is this
correct
similarly we have done here this guy is
X I it was rising on that you add Z K
you get X I plus y Z plus side this part
okay so what I can say that I had this R
vector which is very difficult to
visualize I decompose this into three
components that is X I in this direction
Y Z and Z correct please not so I
decompose this guy into X I Y I say and
and note that this kind of coordinate
system follows my right hand funda so
you move from X to Y your Z Direction
should be your thumb knowledge so Y
decompose this guy into this is one of
the wave you write the vector so now
we'll take one compound or vector where
the length or magnitude of any vector
so here we'll determine the length of
any vector by applying Pythagoras
theorem applies so this is my R I have
to find the length of this the length of
this if you can see is nothing but if
you see L so you this man R vector is X
I plus y Z plus Z key as we have
discussed this part now I have to find
length of on there is nothing but X
square plus y square plus Z root so find
this is not that difficult actually if
you see in this triangle this is y right
and this is X this is y this is X so
this guy will be room of X square plus y
square correct so right here ho P 1 is
nothing but root of X square plus y
square correct right in triangle o t1s
take you see this triangle op1 as this
guy's right triangle o p1 s o p 1 s is a
right triangle
o even s is the right random op1 has
nothing but o s is nothing but my Y and
P 1 s nothing but X so you get root of X
square plus y square now on this if you
see this triangle this triangle this way
is a right triangle here
no I'm talking about this triangle o P 1
so if you talk about P 1 P o P 1 P might
even is a right triangle and my over P
is our op1 I already found and P 1 P is
nothing but Z right so when R is nothing
but or vegetable timber root of op1
square plus z square but is Oh P 1
square X square plus y square because o
P 1 is root X square plus y square 0 P 1
square is X square plus y square n plus
s and that is my magnitude of
like finding this was not that difficult
just I apply this pythor Aslam thrice
twice and I got this formula so these
two formulas are very critical for me
please remember you have any vector R I
can buy it as X I plus Y Z plus Z K
where x y and z are the put the x y and
z currents of this position vector r and
the magnitude of this vector r or you
can say the length of this vector from
this Center 0 is X square plus y square
plus Z let us take some example how to
find the length of these vectors where a
vector a is I plus J plus K so here is X
X is going to 1 Y is going to 1 and said
is equal to 1 so my length of this
vector as I know in other word root of X
square plus y square plus S Pen YX 1
because it is 1 into I 1 into Z 1 into K
so 1 X is 1 Y is 1 right this becomes
root of 1 square X is 1 1 square plus y
square 1 square and there is also 1 2 1
square this becomes root of 3 so the
magnitude of this guy is root 3 similar
in this case my X is 2 2 y is minus 7
and Z is minus the magnitude of BI is
nothing but root of X square plus y
squared residual what is X square what
is y square minus 7 square what is Z
square minus 3 square Q square becomes 4
minus sin square becomes 49 3 square
become 9 so this is nothing but root of
sixty two and that is master so finding
magnitude is very simplify a vector in
the form of X I plus y is a plus Z key
the magnitude is only one to talk X
square plus y square plus X so till now
we have learned what is vector how to
write the vector in the component form
let's do some examples let's get
ourselves more used to vectors
let's write two different vectors having
same magnitude so I can write any two
vectors for example I say if my vector
is in the form X I plus 4a I plus VJ
plus CK my magnitude is root a square
plus B square bustles right if I flip
this a B and C let's suppose it is bi
plus AJ plus CK but the different
vectors here also manage it will be same
only v square plus is purposes both are
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