Examples : Equation of a Plane in Normal Form Video Lecture | Mathematics (Maths) Class 12 - JEE

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three-dimensional geometry part 17 is
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video please make sure that you have
watched part 1 116 this is an example
here you find direction cosine norm of
the plane and the distance from the
origin where my equation of the plane is
given so we have two examples here so
the first is X plus y plus Z is equal to
1 so we compare this equation we have
equation of the plane is LX plus M by
plus NZ is equal to d where L square
plus M square plus n square is Badou
what correct this is my equation of the
plane now if you see my this equation is
X plus y plus so in this case if I
assume n is equal to 1 M is equal to 1 z
is equal to 1 then L square plus M
square plus n square is equal to how
much 1 plus 1 plus 1 3 but I don't want
that I want that my L square plus n
square plus M square has to be 1 so what
I can do is my L square plus n square
plus M square is coming out to be 3 to
get it 1 what I will do is I'll divide
everything by root 3 so actually what
you get here is Direction ratio that is
all I have to find Direction cosine so
I'll divide everything by root 3 now if
you see I get L is equal to m is equal
to n that is 1 by root 3 all the same so
if you say L square M square n square
you add you get 1 only so this is my
actually perfect equation because my
equation has to be in this form LX plus
my plus MZ is equal to B then the
condition that L square plus M square
plus n square is equal to 1 so this was
not the correct form and so I divide
everything by root 3 root 3 now my L
square plus M square plus n square you
compare these two now equation my L is
equal to
M is equal to n is equal to 1 by root 3
there is other relation for science and
distance is 1 by and there is my answer
correct second one also if you have
compared this to X plus 3 by minus Z it
is equal to 5 find the square of this
this becomes u square plus 3 square plus
minus 1 square right 2 so 2 3 1 this
becomes 2 4 plus 9 plus 1 10 that is
root 14 so to convert this into this
form an X plus my plus MZ is equal to D
where L square plus n square by n Square
will do 1 I will divide everything by
root food Turing now you see my L is 2
by root 14 you compare this very LX plus
my MA is 3 by root 14 and as n is minus
1 so now you see L square plus M square
plus n square is equal to 1 now we have
compared this to this becomes D B is
equal to 5 by so just by company you can
see that the direction cosines are these
2 pi out 14 3 we're putting a minus 1
where booty and the distance is 5 by 4
okay let's take few more example on this
same similar question 5y plus 8 is equal
to 0 so this and the other is Z is equal
to 5 plus 8 is because of this thing I
can write as minus 5 y is equal to e Y
because I've tried in this form an X
plus my plus MZ 0 here L is 0 and I said
I know him but M has to be 1 because
root of 0 square root
you're square plus M square plus n
square 0 is equal to 1 M has to be 1 so
divide everything by 5 5 this equation
becomes minus y is equal to 8 by 5 so L
is 0 M because this equation for this
one
my name is minus 1 and is also 0 and B
is C because the equation for cos NX
plus my plus n 0 0 they are L square
plus M square plus n square so only
thing is I am doing isn't converting
this in equation in this form second
I'll take Z is equal to 2 you take this
Z is equal to 2 this is already in the
form these require because the be
required LX plus my plus NZ is equal to
0 L is equal to 0 M is equal to 0 Z n is
equal to 1 and that is correct so we not
convert also so we can say that n is
equal to 1 the direction cosine if you
see Direction cosine will be 0 comma 0
comma 1 in direction distances 2
similarly in this case I should pass
away of 0 minus 1 0 where is I mean not
the difficult question the only thing is
you have to convert this in the form
yeah correct here we are looking for LX
plus my plus n 0 is equal to 0 where L
square plus M square plus n square is
equal to 1 now let's understand this
patient of plain normal form so we take
some example on this find the vector
equation of the plane which is 7 in a
distance of the origin and the normal to
the vector is 3i plus 5 j- just now a
Cartesian form is the vector form only I
don't know why it is written like this
so here if you see the end
there is three I plus Phi J minus six
game right the distance is seven unit
from the ocean the equation we know is
our dot n cap is equal to me please not
we don't say our dot n vectors building
that we don't see always a r dot n cap
that is unit vector of normal so we
define unit vector now but first say if
n is so n cap is nothing but n vector by
magnitude of n vector there is three I
plus Phi J minus 6 K by new table Chi
square plus y square plus minus six
square it was again 3i plus Phi J minus
scale by seventy this is my n cap so my
answer is r dot n cap that is this guy
3i plus Phi J minus K by a hoop seventy
right is equal to D and D is 7 and that
is my answer if you want to take this
guy this seventy this side you can take
that little becomes r dot 3i 8 plus Phi
J minus 6 K is 17 to yourself and as
well please note it is r dot and Kappa
security is not r dot n vectors equality
is wrong correct is r dot n cap is equal
to DS a cap and arrow makes a huge
difference
we'll take one more example you to find
the Cartesian equation for the Sigma for
this particular thing r dot I plus J
minus K is good this is my to find the
Cartesian form the only thing you do you
have to make RS X I plus YJ plus so you
put this value in this and you get the
answer that means if you see my patient
is answers nothing but RS X I plus YJ
plus ZK
dot I plus J plus K is equal to sorry
minus K Delta so all this X into 1 is X
Y into 1 is y 0 minus 1 is minus Z so
converting a vector onto a cart a human
is very simple just make RS XY plus 5
say Plus that can solve it again my more
questions about that we define the
Cartesian of the following equation very
simple nothing great in this just make
RS X I plus YJ plus ZK
you thought of this 2 I 3 J minus 4 K we
put the one you saw this 2 into X
becomes 2 X minus 3 into y because minus
3 y minus 4 into Z is minus 4z again
similar to this you defined a Cartesian
product for this plane here it is
difficult to solve but not that
difficult almost same
I just put our as X I plus YJ plus ZK
heroes I will put right now let's see
how what we get so this is nothing but X
I YJ ZK dot right yes minus 2 T 5 plus t
minus TJ plus 2 s plus T K this is
nothing but s minus 2 T X plus 3 minus T
into y 3 minus T y
to s plus T us list Z is equal to T and
that is my answer simple one
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