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

hello friends this video on
three-dimensional directory part 13 is
rocky by exam Viacom no more free from
exam before watching this video please
make sure that you have watched part 1
to power 12 now we'll go through a very
important concept pulse few lines
what are skew lines skew lines are lines
which are neither parallel nor intersect
so if you see in two-dimensional if we
talk about two dimensional line is
either always parallel or they are
intersecting right you can't have a line
which is neither parallel nor is it
because we are talking only about two
dimensions but when we talk about three
dimensions we can have such lines for
example I have a line like this and this
line example is like this and this line
is like this so this line is in my XY
plane and this guy is in X exactly so if
you see these lines will never meet this
guy's will ever be mean if you extend
this this guys will never me they'll
ever meet you you feel that they are
meeting but actually they will not mean
they're not mean they will be different
altogether right because this is a
different plane is a different plane all
together right so there will never be
such kind of concepts come only in 3d so
in 2d there is no schooling but when you
talk about 3d yes we have skew lines
because three damaged we have more
dimension so we have case where the
miser skewed that is they are neither
paths are not intersected and in three
dimensionally there are two mines and
they are not intersecting it is very
interesting to find a distance between
them
why because in real life you have
everything in three dimension all the
engineering both we do right engineering
work everything in three-dimensional and
that case we actually need to find
distance between them so
when we talk about skew lines we also
talk about distance with this do you
like so let's let us take a very
important topic called distance between
to you skew lines in the vector form so
I have two lines one is good doing even
plus lambda even and odd is gonna do a 2
plus mu 2 V 2 this is r1 this is r2 two
lines I have to find this in between
them the short of this mean that he is
not hard to find the shortest distance
between to do this let's suppose I have
two lines and a one let's suppose this
point is even and this point is a two
right and this line vector will be
lambda even and this will be lambda with
you I have to find short as decimal in
them shorter this Newton then will be
given by this formula even crosby 2.82 -
even vector by even cross v - magnesium
so to prove this let's join this line
even it also let's assume that this is
the sort of this density the clue we
know that a shortest the line which is
the shortest in these two lines that
line we perpendicular to both of these
few lines so let's find the direction of
this because that is a interesting thing
for us because we know the direction of
this right so let's find a unit vector
unit vector and let's assume a
decent-looking them is d and let's give
some name this guy is s this T the speed
is very skewed so first thing I know
that in this case is PQ will be P cubed
and you have to find D vector actually
write D distance D so let's find the
unit vector first and in gap so n cap
will be nothing but cross product of
these two vectors these vector lambda B
lambda we do and then divided by
magnitude of B slit that is lambda
Beaman cross lambda mean by magnitude of
even close nominee sorry mu we do this
is lambda M evil this is new video
correct
so if you see lambda mu cancels so what
I get is my unit vector is nothing but D
1 cross B 2 by - it off even this is my
unit vector that I have got you know
that this PQ vector is nothing but if
you see the PQ vector is nothing but D
into n vector because this is the
distance and this is the magnitude so I
have this is the magnitude good I got
the collective correct now I have to
find this distance D distance D will be
nothing but distance st there is st
distance of st not even a stupid it is
still the best T into cos theta where
theta is nothing but the angle that
suppose st vector angle between st n PQ
that if it is st and let's suppose i
have is there PQ here PQ vector is like
this theta is my angle they both are
same actually
so this st cos theta will be minus guy
what is cos theta actually cos theta is
nothing but my st dot PQ s t vector dot
P Q vector by magnitude of s T and
mention to be my correct because a dot B
is magnitude of a the magnitude of B
into positive the COS theta will be this
two vector dot product by magnitude of
estimation we go STS tickets cancel so
what I get D as st is what a to - even
metal it to - even victim
dot P Q what is P Q vector PQ Victory's
d dot n vector d dot n vector is what
this guy B 1 cross V 2 by V 1 cross B 2
magnitude correct by magnitude of PQ
magnitude of PQ is d by D DD get cancel
and thus I get distance is nothing but a
2 minus even dot B even cross V 2 by
magnitude of D 1 Cross P and this is my
distance between PNG and there are sort
of distance videos so I can say that the
shortest distance between two skew line
is a 2 - even dot B even Crosby - by
magnitude of even cross V du where I
have two lines as even plus lambda V 1
and other line has a 2 + new video let's
do the same thing in the Cartesian form
so I have these two lines a little empty
and I have to say that this is with me
step this is this guy so if you see that
I just convert this guy this is R 1 this
is nothing but line passing through x
y&z X&Y runs at one end and that is
valid to even be one see this is nothing
but X 1 I plus y 1 J cap plus said one
key cap plus lambda into a 1 I plus B 1
J plus C 1 and this guy is nothing but
line passing through s 2 y 2 Z 2 n
parallel to a 2 B 2 C 2 vector that is
x-two i-cap Y J cap has said 2 K cap
plus lambda into lambda in the new or
against the lambda 2 also is lambda 1 is
lambda 2
into a to ICAP plus B to j-cap plus sin
t Union is a two equation and for these
equation I know that the angle between
them is also this the distance is
nothing but a 2 minus even that is this
guy X 2 minus X 1 I plus y 2 minus y 1 G
Plus z2 minus z1 K dot product of this
into cross product of these two even a
tube even even B 1 C 1 a 2 B 2 C 2 this
will come out to be i JK even B 1 C 1 a
2 B 2 C all right matrice dot blood of
this divided by magnitude of B even
cross B that is this then magnitude of
this ijk even B 1 C 1 a 2 B 2 C right
because this guy is nothing but if this
guy's P vector this guy's Q vector this
guy's P cross Q right this guy is P
cross rule and that's what I am looking
for so I am dividing the magnitude of
this so if you solve this
you'll get similar to this if you solve
this you'll get nothing back D as x2
minus x1 you see you get even it this
will get Y 2 minus y 1 v1 we still get
like 2 minus Z 1 C 1 C 2 right this is
the matrix divided by we see the
magnitude of this you see this will come
out to a matrix I into B 1 Z 2 minus B 2
C 1 minus J into a to see a 1 C 2 minus
a 2 C 1 and K into a 1 B 2 minus B 2
signal and that if you take magnitude
this will come out the same thing here
so B 1 C 2 minus B 2 C 1 square plus
this guy C 1 8 minus C 2 a 1 square
Plus even we do - same thing so this is
exactly what you get so if you solve
these so if the lengthy thing it is all
I want to solve it for you you can do it
on you on this means a little bit of
knowledge of matrix and the vectors we
should be already discussed you can
solve this it exactly get same data
advanced so if you have two equation of
line given in the Cartesian form you can
find the distance in this formula or
better is you just convert this into the
vector form line you can use effective
only in that case you don't need to
remember two formulas you can be how you
can solve the question only with one
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