Angle between lines - Introduction to 3D Geometry Video Lecture - Class 11

hello friends this video in
three-dimensions retreat part 6 is lock
you by example a calm no more free from
exam before watching this video please
make sure that you have washed part 12.5
partition form same thing in the
Cartesian form
so we have a plus lambda L is equal to R
vector in the picture form let's try to
loosen the Cartesian so to do this what
I can do is I already know my R vector
is a plus lambda L vector correct since
my R vector is something which is
unknown to me right
so as AR is X I YJ and X K Y it goes
arbitrary something which is unknown to
write a vector is something which I know
so let's a be some point X 1 I plus y 1
j plus X 2 T because this is something
which I know so if I have some a vector
here and this point is x1 y1 z1 this is
something I know right
and I have some Liston line this my R
vector and this is nothing but some
point XYZ in this might some variable
point right and I have some line vector
line with line already to be AI BJ and
see because this line e i VJ CK this is
also I something I know right and x1 y1
z1 I don't because these x1 y1 z1 also
belongs to X Y Z but I knew the value of
this maybe this is 5 6 or something this
value I know so I made x1 y1 then the
normal car make our nation we use you
say expand my 1 x1 x2 y2 the database
you know the value of this you will say
X Y Z that is they are variable they are
so I take any point X Y Z on this line
so equation says R is it going to do k
plus lambda or I say R is nothing but X
I plus y Z plus Z King is equal to a
that is x1 y1 z1 x1 plus y1 plus 31 plus
lambda into L LS m and B once even write
a I plus V J plus C why I do this guy LS
ABC because there is a repent of XY is
that right I just have ABC because the
Valor will be known anyway so this if
you compare these two equation I get X
as X 1 plus lambda it's 1 plus lambda a
Y as Y 1 plus lambda B Y 1 plus lambda
me being small and Z as Z 1 plus lambda
see he said one class that's this I got
by component
correct so from this I can find lambda
lambda lambda so in this case my because
lambda is common so I get lambda is X
minus x1 by is lambda here I get Y minus
y1 by B is lambda and here I get Z minus
Z 1 by CS all are lambdas so equate both
side what I get X minus x1 by a is equal
to Y minus y1 by V is equal to Z minus
z1 by C that is equal to some value that
is my creation of
line in Cartesian form for free-range
correct this is not even required say
more you can put it so this is my
equation of please not so till now we
have not learned this we had not
represented the three-dimensional lining
in Cartesian form and this I have done
this so first we did was we found this
equation of lying the vector form R
equal to e plus lambda L then I told R
is any point in this line so I made R is
X Y Z since right a point was my some
point it please note that this e vector
is not equal to this way they are
different this is the effect of
different a vector or if you are
confused if we put that this P this is P
vector let me print magus B vector so in
this case will be P lector plus lambda L
and this guy will become T plus lambda
and P is X by that yeah this is correct
now this is no confusion so our vector
will be P extra plus my lambda L vector
P is any point in this line and LS the
line is line vector that is talent to
this line so there is a difference
between line vector and line equation
right hope you understand this in this
case my R vector is something which is
denoting line and my line vector is this
line why I am saying R because if you
take any or this this or this it
satisfies this line equation so R is
movable actually you take any point XY
as it here R will satisfy right but you
say line vector is nothing but a vector
that represents the line so that will
not have any movable kind of fun Dada do
you know you take any point in this
Anderson's whether the patient so here
in this case my R vector is the line
equation thing and
by this this a B and C this thing is
just a mind vector
correct so R is going to do P vector
plus lambda L this is a equation of the
line I put R is going to XYZ because my
R vector was just a variable thing right
is this a point in the line a was or
this P was a constant thing a point so I
mean it please x1 y1 z1 and was the line
that is also constant given to me so I
made LS even a iba-jee a CI and then I
use the same equation compared I found
this lambda to be X minus x1 by a 5 and
7 be inside whereas in the scene and
with this I wrote this machine so let's
take some examples where we have a line
that passes to a point invalid using
vector we have to find the equation of a
line is passing through point this and
this palette to this vector so that
means we are given a point this guy is a
point 1 3 and let's suppose this guy is
my producing so this guy is my point
right let suppose this point is a vector
and this line is also parallel to this
line vector that is 3i plus 2 J plus -3
right and this BL rectum so I have to
create a line like this this is my mind
which is passing through this point a
and this garland do this guy
so let's seek some point are in this
soon this was ever this becomes member
you don't need to remember the formula
clear understand the concept we can
marry went awry the formula every time
so in this case magnin our vector is a
vector plus lambda a vector what is a
victim I plus 2j plus 3k I plus 2j last
key K vector plus lambda into L is 3 I
plus 2j minus
and this is nothing but I plus lambda3
sorry 1 plus lambda 3 I plus 2 plus
lambda 2 J plus 3 minus 2 lambda t minus
lambda and this is the equation of the
line very simple we have this point
vector do you have this line vector
right and we have this equation of line
correct why because R is again a
variable you see this is a line vector
this is a point vector you can have any
are any odd it's a variable right all
these are are they're all these are
patient of the line so this is an
equation of line where R is a vector
plus lambda of it if you see this guy is
also a vector R vector and this vector
represents equation of line because this
is a variable vector actually you take
any then move this correct let us take
one more example we refine the equation
line in the vector in Cartesian form
both form in this case earlier it was
only vector form the passes through this
position vector and it is in this
direction that means this guy is this
guy is 2i minus J plus 4t right so this
guy is my position vector legible this
guy is e vector and this is parallel to
I plus 2j minus here correct this be a
vector so to find equation line that is
passing through this and collar this
this be our victim and this becomes
lambda into L vector so my R vector is
nothing but a plus lambda L vector and
this is nothing but is 2y minus J plus
4k plus lambda all the caps action
lambda L vector there is I cap to J cap
minus K cap I cap plus TG cap minus K
can correct this is my fish in the line
that is nothing but 2 plus lambda I plus
it will be 2 lambda minus 1 J cap plus 4
minus lambda J cap this is my patient of
life correct so if you want to and this
is nothing but R and R I can write as X
I plus y J cap plus Z key cap so this is
my equation of the line in vector form
and find in Cartesian form so I just
wrote R is equal to X I plus Phi Z plus
ZK compare both these two so I get X is
equal to 2 plus lambda I am finding
lambda in both all that is compared with
lambda or and this is equal to Y is
equal to lambda minus 1 and Z equal to 4
minus lambda so with this I get lambda
is X minus 2 with this I get lambda is y
plus 1 by 2 now this I get the lambda is
0 minus sorry for menu and all are equal
to lambda simply create this so I get X
minus 2 by 1 is equal to y plus 1 by 2
is equal to 4 minus XL is side Z minus 4
by - all right so this is nothing but
minus off
this is my equation in Cartesian really
a good thing I have this point I have
this line I wrote this equation I'll go
to E Plus lambda L I got this equation
in vector form and to convert this into
a Cartesian form
I just would are as X I plus YJ plus ZK
because R as any point
I know XYZ on this line where X Y Z
since Phi this line equation and with
this I can
we hit these two guys to get lambdas
right all this lambda equals I wrote
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