BAM! Mr. Tarrou. In this video we're gonna do a single example of lines in three space
I'm laughing because I hardly ever do a single example in a video we have two lines that are
in Vector form and we're going to determine or show verify that these two lines are indeed skew
so how are we going to do that well the idea of being lines being skew in three space is that we
have lines that do not intersect now on two space the only way that two lines don't intersect is
if they're parallel and they don't maybe say have the same y-intercept where there's like
many solutions we want no Solutions so in two space we only have to have parallel lines that
say have a different initial point or a different y-intercept but in three space you can have lines
that are parallel or you can have lines that are they're not parallel they're not their gradients
are different but they still do not intersect you can see that on in a room now you can't see the
walls in my room but like the line which is where the wall and the floor meet uh doesn't intersect
with a line where the wall and the roof meets goes in a different direction so skew so we have
two conditions that we need to check not just one the first time I saw this question I just looked
at you know do they intersect and just ignore the whole idea of them being parallel so here
we go for these two lines and we do have two lines in Vector form which means that we're looking at
um sort of like a vector R is equal to Vector a plus Lambda or beta or alpha or whatever Greek
letter we have that is a scalar multiple of a direction vector or gradient vector
so I might go in between saying coordinate and position vectors so so because Vector R would
be like the position Vector from the origin to coordinate R and Vector a is a position Vector
from the origin so like o capital A so like a will have a coordinate and that's going to be a
position Vector for an initial or a known known point that the line contains and then we have
that plus Lambda times some kind of gradient or Direction Vector so I might say that this is the
coordinate of a point but it's really a position Vector to get to that coordinate in three space
so just be aware of that sometimes I kind of go back and forth a little bit uh arbitrarily
so in other words when you're looking at the equation of these two lines in three space you are
seeing with this first vector and component form of one two two the coordinates basically the x y z
coordinate of a point that line contains and then after the scalar multiple of some kind of Greek
letter usually of in this case we have Lambda and mu we have another Vector here of negative
one one and two well this is the direction Vector sort of the slope if you will the gradient Vector
of line one and line two contains the coordinate then the initial point of one two four and we have
a direction Vector of 2 1 and 6. well we just you may just be learning vectors right so we
have vectors that are going in the same direction and then that's fine but you can also with the
idea of parallel vectors those vectors can be going in opposite directions but have the same
basically the same gradient but maybe with just a multiple a scalar multiple of negative one so
we're going to do a parallel check are these two lines in three space parallel well if they're
parallel then they're arguably sort of like not skew they may not intersect with that different
initial coordinate but for the direction vectors that are defining the direction the slopes of
these lines if they are parallel parallel vectors are scalar multiples of themselves so that means
that we have or we're going to check is negative 1 1 2 a scalar multiple of the vector 216.
so what that means is basically we need to see you know we can see that is K going to
be the same scalar multiple to convert this coordinate or movement along the x-axis into
this movement along the x-axis and such is also one times something going to be equal to one this
movement along the y-axis is a is it a scalar multiple of this movement along the y-axis and
then two k and 6 the movement along the z-axis so looking here we have negative one is equal to
K times 2 which is going to mean after we divide by 2 that we have negative one half is equal to K
so that is the scalar multiple to look along the movement along the x-axis if we do the same thing
with the Y's we have this y movement is equal to K giving then multiplied to this movement
along the y axis of one and here we get a scalar multiple of one which is equal to K well negative
one-half is not equal to one so therefore these two gradient vectors are not parallel
great so these two lines are not parallel that's half the check if you want to show that these two
lines are skewed so are they not parallel and not intersecting so for that to be the case
just think about um you know saying we have line of Y is equal to you know 2x minus 3 and
Y is equal to negative 1 X plus seven like maybe an algebra 1 problem you know Finding where these
two lines intersect you would do that at least in the way that I've given this to us we would
do that through substitution and you'd plug information about the second line into the
first and start solving start finding that point of intersection well what that looks like with
lines in Vector form that are in three space you're going to want to find the component
forms the parametric form excuse me of each of these lines and start trying to figure out what
the scalar multiple is there and if it's the same if they intersect so for line number one
we have any new X is going to be found
by starting with our initial x coordinate of one and adding with it Lambda times negative one
and Y any new y coordinate that is on the first line is going to be found by doing our
initial y coordinate of 2 and then plus Lambda times positive one and every new Z coordinate
is going to be found by taking the original Z coordinate up to and
adding it times Lambda times 2 that's for line 1. now for line two we have
X is equal to going into parametric form the x coordinate of any point along the line is going
to be found by doing one plus two mu it's going to put the coefficient first like it kind of normally
would be written our new Y coordinates are going to be equal to 2 plus mu and Z is equal to uh
4 plus 6 mu so let's start substituting you know bringing together the information that we know
in parametric form to these two lines just like substitution with algebra 1. let's bring together
D x's and if x is equal to one plus negative Lambda and X is equal to one plus two Lambda
well those x coordinates would have to be equal if these lines are intersecting so that's going
to give us 1 minus this is for the X's 1 minus Lambda is equal to 1 plus 2 mu meaning that
we have subtracting both sides by one we have negative Lambda is equal to 2 mu and mu or Lambda
then is equal to negative 2 mu doing that with the Y's bringing these together we have 2 plus Lambda
is equal to 2 plus mu
which means Lambda is equal to mu so we're trying to now you know we're trying to solve
the system of equations and we have two equations here and two unknowns we have
Lambda is equal to negative two mu and we have Lambda is equal to Mu if we bring these
together through Lambda we have negative two mu is equal to Mu which then gives us sort of
like if we divide both Supply both sides by mu we have negative two which is equal to one so
okay that's kind of weird that doesn't really work so really the only way that these two equations
can be true at the same time is if Lambda and mu are both equal to zero so we've got now
yet the only Lambda and mu that make these true at the same time is if
mu is equal to zero and Lambda is equal to zero so can we put now remember this is three space not
two space and I've only matched up the movement along the x-axis and y axis so right now we're
kind of like in two space we need to look at three space with these values being the only values that
create truth between these two equations we plug in you know zero equals negative two times zero
zero equals zero great and zero equals zero great then we need to be able to plug these values of mu
and Lambda into our parametric equations for Z which is z is negative two plus two Lambda
let's pick a different color
well this is asking for Lambda and we've determined the only answer could be zero the only
number we could use is zero and over here at Z we have four plus six mu so four plus six mu well
by setting the component forms of the parametric forms of X and Y together that means that mu has
to also be zero and when I do that I get 2 is equal to 4 for Z and that's a false statement
now I had that sort of weird statement as I tried to bring together the system of equations for X
and Y but I had mu and Lambda open in those equations and say okay well this is bringing
these together through substitution gives sort of what seems to be a false answer unless both
of these variables are zero so we're kind of stuck there but now using the only possible
values of mu and Lambda that we could come up with plugging those into Z still gives us
a false statement so therefore these lines do not intersect line one does not intersect line two and
line one and line two aren't parallel so thus we have shown I probably should write this out in a
sentence but so we have shown in the two checks that these two lines are indeed skew they're not
parallel and they do not intersect so they have to be skewed I'm Mr true bam go do your homework