## (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,