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engineering HSC and IIT JEE Main and
advance video Rectangular vector analysis
topic named Rectangular coordinate
system Hello friends in the last lecture
we have seen the difference between
scalar and vector simple vector
operations now electromagnetic field
theory we have the vectors of electric
and magnetic field
so to give exact location position to
the vector in three dimensional space we
require certain system what is called as
coordinate system in our syllabus we are
going to go through the three coordinate
system types rectangular which is also
called as Cartesian the second book will
be cylindrical and the third one
spiracle coordinate system the Muto's of
all the coordinate systems are same but
the rules are different so let us begin
with the first coordinate system
rectangular coordinate system which is
also called as Cartesian coordinate
system
coordinate systems are of three types
Kardashian also called rectangular
coordinate system
second one cylindrical coordinate system
third one spherical coordinate system
the basic motto of coordinate systems is
same to give a particular position or
location of a point or we have done into
the three-dimensional space but the
waves which these three different
coordinate systems are different work is
same the rules are different in
Cartesian also called rectangular
coordinate system suppose a point P we
have to locate the coordinates are
represented by
capital P in bracket X Y is here XY is
here
use all the majors for three distances
covered by the point P in the
three-dimensional space along x axis y
axis and z axis whereas the same point P
is represented in cylindrical coordinate
system as Rho Phi ship here Rho and said
are the distances where as Phi is the
angle the same point P if we have to
locate in spherical coordinate system it
is represented P in bracket R Theta Phi
if we switch from rectangular coordinate
system to cylindrical coordinate system
one coordinate we have common that is
smalls head when we switch from
cylindrical coordinate system to
spherical coordinate system one
parameter one coordinate we have same
that is Phi K now in critical coordinate
system all the three coordinates are
measures of lengths or distances in
meters in cylindrical coordinate system
to coordinates Rho and said are the lens
and Phi is the angle ile spherical
coordinate system there it is the road
of two angles theta and Phi and R is the
measure of length as the names of these
coordinate systems described in the
rectangular shape the cylindrical shape
and spherical shape here X Y and C
describe the length breadth and height
of a rectangular our terminal people in
cylindrical coordinate system Rho
defines radius of the cylindrical shape
whereas R in spherical coordinate system
describes the radius of spiracle shape
so let us start with first coordinate
system called
Cartesian or rectangular coordinate
systems now rectangular are also called
as Cartesian coordinate system first of
all we will try to represent a point in
the three-dimensional space so for
locating or positioning a point into the
3d space we require certain references
with respect to which the exact location
position has to be calculated so in the
rectangular also called Cartesian
coordinate system there are three axes
represented capital X capital y and
capital shape which are mutually
perpendicular to each other which we
also call orthogonal to each other
so in the diagram here it is x axis y
axis and z axis that are perpendicular
to each other
intersecting in the origin oh yes is
perpendicular to Y Y is perpendicular to
say and say it is also perpendicular to
the X now in this diagram I have drawn a
excelsis y axis and z axis in the plane
of the paper okay actually these x axis
is one sided orientation okay so this is
positive side the negative side of x
axis can be drawn in opposite direction
this is minus x y axis in this direction
has positive orientation the negative
side can be shown by a dashed line this
is minus y the third axis this has
positive side the negative side can be
shown like this
- in this diagram it will be more clear
that the two axis forms a plane suppose
this is positive side of y
this is positive side of X this is
positive side of Z so x axis x axis
ranges from like this from minus
infinity to plus infinity
y axis from minus infinity to plus
infinity so x axis and y axis forms this
plane and C axis is perpendicular to
that plane this plane you can consider a
horizontal plane so here if you talk
about the plane of the paper so Y axis
is towards the right side of the plane
on the paper in positive negative
towards the left z axis towards this one
in positive downwards negative x axis is
actually perpendicular to the plane of
the paper so these are the three
references mutually orthogonal or
mutually perpendicular to each other in
another words the references can be
taught as XY plane Y is head of length
and XZ plane let us see in this diagram
if this is x axis this is y axis and
this one said axis so this plane is XY
plane now Y Z plane is represented like
this
and X Y plane is represented like this
okay the behind portion can be imagined
okay so these three planes intersect
again in a single point what we call as
origin okay so either of the references
are XYZ axis represented in
CAPITAL LETTERS or XY wise head and exit
plains these three planes are also
mutually perpendicular to each other
making 90-degree and intersecting into
the point O now in this
three-dimensional space if you want to
represent any point P let us take this
is the point P we have to give three
coordinates small X small Y and small
shit so what are these small XY ends
here if this is the x axis this point P
has covered a distance along these
capital x axis
so this which is X what is this Y it is
the distance covered by point P along
the capital y axis so this is y from
this point to this point and what it
said it said if you drop a perpendicular
on to the z axis how much height is
attained by that particular point on
this head axis from the origin this is
nothing but shit in another way point P
is here we draw a perpendicular on the
XY plane which will intersect here let
us say this is P - from the P - we draw
a line parallel to Y which will cut else
here so this is small X x coordinate a
line parallel to x axis cut why here so
this is small Y dimension and if you
drop a perpendicular onto this head axis
the height attained by the point P from
the XY plane or from the origin is said
okay so if you join the origin to this P
- P - is in the XY plane
so this line and this line are parallel
ok so these are the measures for length
x y and z this diagram also gives you
the 3d visualisations of three planes XY
y is
and intersecting into the origin so
these are supposed to be the references
with respect to which we take the
measurements of the point P in the
three-dimensional space now in the
rectangular coordinate system we have
seen how to locate a point P having the
coordinates XY and Z so these small x
y&z can have any constant or real number
values may be positive or negative
okay so these are the distances covered
by the point P in three-dimensional
space from the origin with respect to
capital X Files hid access now this is
the structure I have drawn I will always
draw X X is oriented like this Y axis
oriented like this and same axis
oriented like this this I called the
orthogonal structure as perpendicular to
each other along with they are seen to
be the right handed coordinate system so
what is a right-handed coordinate system
if we go from x axis to Y axis with
screamo okay so the screw will be moved
forward into the same direction or
right-hand thumb rule if you go from X
to Y we get inserted so it should be
positive if you go from Y to Z we get X
so X should also be positive or two else
it should be Wow so the representation
can be like this X to Y use said Y to Z
gives X and Z to X gives us Y so these
are perpendicular to each other this is
called as right-handed coordinate system
now let us talk about any planes we can
represent in rectangular coordinate
system the planes we can represent small
X is equal to constant small Y is equal
to constant small
so it is equal to constant X is equal to
constant Y is equal to constraint C is
equal to constant before putting any
constant value if we put X is equal to
zero what is the case so it becomes a
criteria that in the three-dimensional
space we have to collect all such points
which are satisfying the criteria
yes is equal to zero so in the three
dimensional space X is equal to zero
this is the plane of the paper from left
to right I have y-axis from bottom to
top
I have zero axis and x axis is actually
perpendicular to it now I put the
criteria X is equal to zero so I cannot
be having the value of x more than 0 or
less than 0 so I cannot be pinning this
paper I cannot be above these paper I
have to be on this paper because on
these paper one dimension is y so it can
have minus infinity to plus infinity
this here can have minus infinity to
plus infinity but X is equal to 0 if we
put X is equal to 1 I am here at the
height of 1 meter or 1 unit you see if
we put X is equal to 2 I have to
increase the height if we put X is equal
to 3 again I have to increase the height
so X is equal to 0 is nothing but Y is a
plane now if I take X is equal to 1 so
here I have to take such a plane which
is 1 meter above or one unit above the Y
Z plane okay so this is X is equal to 1
plane if X is equal to 3 so that plane
will be above the Y Z plane or parallel
to Y Z
at a distance of three units so it can
be drawn like this suppose this is a
plain infinite in this direction this
direction this direction and this
direction and intercepts X here so
capital X exists so from the origin to
this this is the distance X so X is
equal to constant so this is the plane
which is parallel to Y Z plane at a
distance constant now if you take Y is
equal to pastern so first of all we put
Y is equal to zero
why is equal to 0 so Y is equal to 0
means we cannot be on the positive side
of wine we cannot be on the get you set
up why we should be aligned to the
position of Parisian ok X can be
anything on positive or negative so it
can be anything in upward or downward
direction so 1 is equal to 0 is nothing
but X instead of gram episode a plane
which is perpendicular to the y axis if
we put y is equal to 2 so we have to
take such a plane which will intersect y
axis in two units so suppose this is two
units from the origin along the Y so
this is the plane which will intersect
this Y axis in two so this plane is
parallel to XY t suppose I take Y is
equal to minus 2 so on the negative side
of the why I have to put in that
particular plane which will intersect in
minus two Y is equal to minus two so
again it is a plane parallel to Z plane
ok which will intersect y axis at -2
similarly 0 is equal to past and we can
put so say it equal to constant will be
a plane parallel to XY plane
if it has positive value it will be
above the origin if it has negative
values it will be below the origin
now let us say any arbitrary point he is
given with the specific coordinates to
us and we are supposed to represent it
in the three-dimensional space using XYZ
axis so let us see in the diagram he is
given the coordinates 2 3 & 4 it to be
this small X is having value 2 small Y
is having value 3 and small C R is
having value 4
so I have to represent this point a in
this three dimensional space using
rectangular coordinate system now M is
equal to 2 so first of all on x axis I
will mark 1 2 so this is X is equal to 2
y is equal to 3 so on y axis I have to
mark 1 2 & 3 units so it is equal to 4 I
have to mark 1 2 3 & 4 units so here we
have marked X is equal to 2 y is equal
to 3 and Z is equal to 4 so from X is
equal to 2 we will draw a line parallel
to Y axis from y is equal to 3 we will
draw a line parallel to x axis this is
the intersection point so this point has
lies in the XY plane I can represent
suppose P - so it has x-coordinate 2
units y-coordinates 3 units and same
coordinate as it lies into the XY plane
it means Z is equal to 0 so P dash has
coordinates 2 3 0 now we want the
coordinates a 2 3 4 so 2 3 we have but
we want the Z coordinate to be equal to
4 so let us raise the height of this
point P dash by 4 units so
so here suppose one unit 2 unit 3 unit
for unit okay so this two points are at
the same height that is equal to 4 so
this is your point a having coordinates
2 3 & 4
now in terms of vector if you have to
represent t-bar so a bar or a bar is one
and the same representation so here the
wave table in the diagram can be
represented
starting at O ending at a so this is ava
or Oh a bar okay so this is starting
point of the vector o and ending point
of the vector a so who has the
coordinates 0 0 0 and a is having the
coordinates 2 3 4 so for representation
of a vector we always do endpoint
coordinates
- start point coordinates so here we
will have 2 - 0 a square 3 - 0 a YK 4 -
0 is your K now what are these a X cap a
Y cap and either cap these are for DES
unit vectors along capital X capital y
and capital ship directions you need
waved us the word unit vectors means the
wave tar is having unity magnitude unity
magnitude means magnitude equal to one
so it does represent only the direction
of the vector so EF scale represents
direction of x axis so one unit here we
can represent a X cap one unit along Y
axis we can represent a y cap and one
unit along z axis we can represent is
your cap so here I can show you here
this is y YK this is generally four unit
vectors small a is specified the suffix
gives you the particular direction of
the axis whether it may be x y and z and
as we know that x y and z axes are
perpendicular to each other these unit
vectors ax cap a y cap is it cap these
are also perpendicular to each other now
the vector representation using
rectangular coordinate system in general
we can show like this a VAR ax KF k KY a
YK
is it cap suppose another vector we have
b bar so it can be represented VX a of
square dy k YK plus B's head is it cap
so here a is cap e YK is a gap in both
the cases are same these are the unit
vectors aligned along x axis y axis and
z axis now ax
capital a capital a why capital H a
capital B circus xby he said these are
the coefficients of this vector or this
is component vector B along x axis
component of vector B or of Y axis
component of vector B along z axis
similarly for vector K also for a and B
if in bracket we specify some
coordinates let us say 1 2 3 K 4 5 6 so
1 2 3 these are nothing but the values
of x y and z so in a vector
representation we have to take capital K
X means x coordinate equal to 1 of a y
coordinate to a a wife so ax will be
having 1 a Y will be having 2 and a sir
will be having 3 similarly for the be
having coordinates 4 5 6 bx will get the
value for B Y will get value for you be
easier we'll get the value 6 so a bar
will be represented here square or 1 a
of cap plus twice a Y cap plus 3 times
here cap B bar will be represented for
ax cap plus Phi u here Y cap plus 6 is
head cap now these are the
representation of vectors using
rectangular or a Cartesian coordinate
system now let us talk about dot product
dot product is the multiplication of the
two vectors earlier we have seen
addition and subtraction of the two
vectors let us talk about the dot
product which is multiplication of two
vectors for example a bar and B bar now
dot product also called as scalar
product
for vector a and vector B having
representations the dot product is
scalar multiplication
the outcome is multiplication so a bar
and B bar are the two vectors and dot
issue in between the two the formula for
dot product is magnitude of a bar into
magnitude of B bar into cosine of
smaller angle theta between them so in
the space if this is K bar and this is B
bar so if you take a bar and B bar to
have same source point so this is
smaller angle theta between them the
bigger angle theta may also be like this
but according to the definition we
should always take the smaller angle
theta between a bar and B bar so
magnitude of a bar into magnitude of B /
into cosine of smaller angle theta so
this operation use us magnitude only it
does not have any kind of direction in
between so it is also said to be a
scalar product now how to find magnitude
of a bar or B bar so if B bar is having
the representation a of ax cap a why am
i cap plus is it easier to cap so if a
why is your capitals represent the
components along XYZ axis or coordinates
simply you can say for B bar we have V X
K XK be y a YK plus B's here is your cap
so magnitude of a bar is given as
under root K by Del X square plus
capital e Y square must capital case
your square magnitude of B bar is given
as capital B F square capital B Y square
ok put a be Z square the coefficients K
of K by is e coefficients be V X V Y V Z
have to be squared and added it and
finally the square root has to be taken
this gives us the Magneto
this is actually from the distance
formula in the three dimensional space
suppose
sex y and z-axes are there intersecting
into the origins and the point P is
there so if P is there in the space
having the coordinates let us say x1 y1
z1 another point in the space let us say
Q having the coordinates x2 y2 z2 are
there if you want to find the distance
between the point P and Q then we have
to write more of PQ bar so what is P Q
Bar
P Q Bar is a vector starting at P ending
at Q so I can show it like this this
victim started at Point P and ended at
Point Q the magnitude of that is nothing
but the distance between these two okay
so this is distance okay so here first
of all P bar as a vector we can take
which is nothing but origin to point P Q
Bar as a vector we can take which is
from origin to Q so here if you see this
is P bar ROP bar this is o Q Bar or
simply Q Bar these are the two vectors
okay by the triangle law of addition we
can see Oh P bar plus P Q Bar is equal
to o Q Bar okay so simply P Q Bar K P Q
Bar is nothing but the oak cuba - o
people o Q Bar minus o P bar so ho Cuba
with these coordinates x-two y-two said
to kill o P bar with the coordinates
x-one y-one said well we can simply
write the distance formula under root of
x2 minus x1 mod square sorry letter
square Y 2 minus y 1 square plus
let's a 2-0 square so this is the
distance formula we have seen dot
product which is called as scalar
product of the two vectors represented
by Ava Rotem Biba for example so a mod
of B bar is nothing but magnitude of
Kaiba into magnitude of B bar into
cosine of smaller angle theta between
them
so here k bar b bar and angle theta we
have discuss now cross product the cross
product is represented a ba cross of b
bar the outcome is a vector for the dot
product
the outcome is having only magnitude
that is Killa
hence it is also called a scalar product
whereas for cross product the outcome is
vector generally it is calculated by
using the determiner the first row
having the unit vectors yes cap a y hat
head cap and the second and third row
corresponds to these coefficients of
vector a bar and B bar so this is
capital S capital a by capital a capital
B X capital B y capital B easy solving
this determinant okay we'll give you the
cross product of k bar cross of b bar
diagrammatically each a bar and B bar
are aligned to have same source points
angle theta with with them a bar cross
so P bar we have to proceed from a bar
to P bar with the curled fingers the
thumb will give you the direction of the
resultant so if we get a bar to P bar
the thumb is inside the plane of the
paper which is the direction so I write
in word or a bar b bar if we have to
take b bar cross of a bar
so be markers of a bar so this will be
outward or Beaver crosswalk Ava the
subsequent lecture will follow with the
numericals based on to the concepts we
learned in the rectangular coordinate
system
thank you
