(12) Finding vector
equations of planes
One way of doing this is to use a very similar method to the one for
finding the vector equation of a line.
The difference is that now we want an equation which gives the position
vector of any point in a flat surface or plane.
It's easiest to explain how this works by starting with the case where the
origin lies in the plane.
The drawing below shows part of a plane like this.
To find the position vector of any point P, we have to know 2 non-parallel
vectors which lie in the plane.
I have called these s and t. It is then possible to get to P
by adding together suitable multiples of s and t.
This gives us the equation of the plane as
r = as + bt.
In my picture, a = 1.4 and b = 1.1 approximately.
There are some more examples of vectors being described in this kind of
way in the earlier
section using components to describe vectors.
Now suppose we have a plane which doesn't pass through the origin.
The drawing below shows part of a plane like this.
Again, s and t are known vectors which lie in the plane.
We also now need a way of getting to the plane from the origin, so we have
to know the position vector of some particular point in the plane.
In my drawing, this point is M with position vector m.
Once we have reached the plane, we can find the position of any general
point P relative to M in the same way that we did above by saying
that p = as + bt.
(For my particular P, a = 1.2 and b = 1.)
Now we can get the equation of the plane in terms of the known
vectors m, s and t.
We have r = m + p
but p = as + bt
so r = m + as + bt.
In the special case where the plane passes through the origin, we can
leave out the m because it is the zero vector.
If two planes are parallel, then the same s and t can be
used for both of them, since we can move these free vectors so that they lie
in either plane. The equations of the planes are different because each one
must also include a position vector from the origin to a known point in
that particular plane.
Writing the equation of a plane in this way has one big disadvantage. There
are infinitely many directions which vectors lying in the plane can have.
Therefore there are infinitely many pairs like s and t to
choose from.
It would be much nicer if we could use a direction which is unique to the
plane.
There is a direction which has exactly this property.
It's worth thinking what this direction is for yourself
before finding out.
Or back to the previous section,
or back to the vectors homepage.