In the first section of this chapter we saw a couple of
equations of planes. However, none of
those equations had three variables in them and were really extensions of
graphs that we could look at in two dimensions.
We would like a more general equation for planes.
So, let’s start by assuming that we know a point that is on
the plane, . Let’s also suppose that we have a vector that
is orthogonal (perpendicular) to the plane, . This vector is called the normal vector. Now, assume that is any point in the plane. Finally, since we are going to be working
with vectors initially we’ll let and be the position vectors for P0 and P respectively.
Here is a sketch of all these vectors.
Notice that we added in the vector which will lie completely in the plane. Also notice that we put the normal vector on
the plane, but there is actually no reason to expect this to be the case. We put it here to illustrate the point. It is completely possible that the normal
vector does not touch the plane in any way.
Now, because is orthogonal to the plane, it’s also
orthogonal to any vector that lies in the plane. In particular it’s orthogonal to . Recall from the Dot Product section that two orthogonal
vectors will have a dot product of zero.
In other words,
This is called the vector
equation of the plane.
A slightly more useful form of the equations is as
follows. Start with the first form of
the vector equation and write down a vector for the difference.
Now, actually compute the dot product to get,
This is called the scalar
equation of plane. Often this will
be written as,
This second form is often how we are given equations of
planes. Notice that if we are given the
equation of a plane in this form we can quickly get a normal vector for the
plane. A normal vector is,
Let’s work a couple of examples.
Example 1 Determine
the equation of the plane that contains the points ,
In order to write down the equation of plane we need a
point (we’ve got three so we’re cool there) and a normal vector. We need to find a normal vector. Recall however, that we saw how to do this
in the Cross Product
We can form the following two vectors from the given
These two vectors will lie completely in the plane since
we formed them from points that were in the plane. Notice as well that there are many possible
vectors to use here, we just chose two of the possibilities.
Now, we know that the cross product of two vectors will be
orthogonal to both of these vectors.
Since both of these are in the plane any vector that is orthogonal to
both of these will also be orthogonal to the plane. Therefore, we can use the cross product as
the normal vector.
The equation of the plane is then,
We used P for
the point, but could have used any of the three points.
Example 2 Determine
if the plane given by and the line given by are orthogonal, parallel or neither.
This is not as difficult a problem as it may at first
appear to be. We can pick off a vector
that is normal to the plane. This is . We can also get a vector that is parallel
to the line. This is .
Now, if these two vectors are parallel then the line and
the plane will be orthogonal. If you
think about it this makes some sense.
If and are parallel, then is orthogonal to the plane, but is also parallel to the line. So, if the two vectors are parallel the
line and plane will be orthogonal.
Let’s check this.
So, the vectors aren’t parallel and so the plane and the
line are not orthogonal.
Now, let’s check to see if the plane and line are
parallel. If the line is parallel to
the plane then any vector parallel to the line will be orthogonal to the
normal vector of the plane. In other
words, if and are orthogonal then the line and the plane
will be parallel.
Let’s check this.
The two vectors aren’t orthogonal and so the line and
plane aren’t parallel.
So, the line and the plane are neither orthogonal nor