Welcome to a lesson on graphing planes to graph planes by hand it is often
helpful to make traces were the plane intersects
each coordinate plane so we'll take a look at where it intersects the xy plane
yz plane and xz plane these are called traces
so to determine the xy trace we'll set z equal to zero
to find the yz trace we'll set x equal to zero
and to find the xz trace we'll set y equal to zero
then what will do is well combine these three traces on to the xyz coordinate system
to make a graph of the plane
now if an equation a plane is missing a variable the plane will be parallel to
the axis
represented by the missing variable let's go and take a look at a couple
of examples
now I should mention we could just go ahead and plot the intercepts
on the three-dimensional coordinate system but since you will be graphing
quadric surfaces very soon
I wanted to introduce this technique now because well use it again
for a quadric surfaces so to determine the
xy trace or where this plane would
intersect the xy coordinate plane
we're going to set z equal to zero and that will form the line
x minus two y equals six and graph this line in standard form and go ahead and find the
intercepts if we set y equal to zero and x would be equal to six
that would be our x intercept
if we set x equal to zero, y would have to be negative three
that would be our y-intercept so this plane cuts the xy plane
along this line here, to determine the yz trace or
where the plane intersects the yz plane
will set x equal to zero, that will give us the line
negative two y plus three z equals six
notice how we have changed the orientation of the axis-es this is now the
y-axis
this is now the z-axis so if we set z equal to zero, y would have to be negative three
now this is the y-axis here so we'll be over here
three units notice the y intercept is the same
in the xz trace as it is in the xy trace
now if we set y equal to zero
z would have to be equal to two so are
z intercept is positive two so the given plane intersects the yz plane
along this line and then lastly to determine the
xz trace we set y equal to zero well if y is zero we will have the line
x plus three z equals six
so are x intercept will be six again
and the z intercept if x is zero would have to be two
so this plane intersects the
xz plane along this line here
so now we have all the intercepts we can go ahead and plot them on the three d coordinate
system
to sketch this plane so we can see the x intercept will be positive six so we'll go out
six units along the
x-axis , pointy ears on the plane
the y-intercept is negative three
so we'll go to the left three units, this was six
this point is on the plane and then lastly the z intercept is positive
two so we'll go up two units
and now well form a triangle with these three points
now well go ahead and shade this, this gives us an idea how that plane looks
in three dimensions showing where it crosses
each axis as well as the
xy yz and xt traces
now it is a little bit challenging to give a three-dimensional perspective
on a flat surface as we see here on the screen
but this is pretty much the best we can do by hand
let's go and take a look at one more
here we have three x plus two y minus six z plus nine equals zero
so determined the x y trace we'll set z equal to zero
that'll give us the line three x
plus two y plus nine
equal zero I'm going to go ahead and move the nine over the right side so this will equal negative nine
so if we want to find the x intercept will set y equal to zero
so the x intercept will be negative three
and if we set x equal to zero we'd have two y equals
negative nine to divide both sides by two we'll have y equals negative four point five
so somewhere down in here
so the given plane intersects the xy plane along this line here
to determine the yz trace will set x equal to zero
so we'll have the line two y minus six
z again i'll move the constant to the right side to have negative nine
so if we want to find the y intercept will set z equal to zero
so we'll have two y equals negative nine we already found that on the other graph
so we know the y-intercept is negative four point five
here's the y-axis in this trace so we're back
here at negative four point five
to find the z intercept we set y equal to 0
so we have to to divide both sides by negative six
negative nine divided by negative six is gonna give us a positive
one point five
so the given plane intersects the yz plane
along this line here
and then lastly to find the xz trace we set y equal to zero
so we'll have the line three x minus six z
equals negative nine
again setting z equal to zero the x intercept would be negative three
which matches what we found up here
and then setting x equal to zero z would be positive one point five
so we've shown where the plane intersects the xy plane
the yz plane as well as the xz plane
now we'll go ahead and sketch it on the three-dimensional coordinate system
so again here's your x, y and z axis
so the x intercept is negative three so we'll go back three units
this point is on the plane the y-intercept
is negative four point five so we'll go left
four-and-a-half units
somewhere in here
and then lastly the z intercept is positive one point five
and then i'm going to go ahead and change the scale on this so we can make a nice triangle
we'll call this one and we'll call this two so here's one point five
will go and form a triangle using these three points
the sides of the triangle represented our traces and to go ahead and shade this
triangle
misrepresent where the given plane intersects each axis
okay that's gonna do it for this video hope you found this helpful