Embedded Files
Equations for lines in two dimensions can be generalised to equations for planes
Equations for lines in two dimensions can be generalised to equations for planes
Imagine looking straight down at a page with x-y axes on it. In three dimensions, we can represent a surface by giving the height of the surface at every (x,y) point.
This could be with a function of two variables, represented by z = f (x,y). This gives points (x,y,z) on the surface.
A real-world example is a topographical map, showing the elevation (height above sea level).
Topographical map showing heights in the Pukenui Forest.
For a plane in 3 dimensions, it can be represented with the equation z = Ax + By + D .
This gives a rule for the height of the plane at any point.
There is an equivalent to the y-axis intercept (0,c) of the line y = mx + c.
When x = 0 and y = 0, we get the point (0,0,D) on the z-axis.
Although this is the best way to understand the equation for a plane, it is not always the best way to represent a plane.
A general equation for a plane is in the form Ax + By + Cz = D .
For example, the plane z = 3x + 2y - 3 can be rewritten as 3x + 2y - z = 3 .
The general equation is not unique, but can only vary by multiplying through by a constant.
For example, the plane above could be represented as 6x + 4y - 2z = 6.
Parallel Planes
Parallel Planes
Think about what would happen if I changed the plane z = 3x + 2y - 3 into the plane z = 3x + 2y + 4 .
At any (x,y) point, the z-value is increased by 7.
(Not convinced? Find the z-values at the points (1,0), at (5,1) and at (3,7)).
These two planes are parallel - they can never intersect, because the vertical separation is always 7.
In general, if two planes can be rewritten in the forms
Ax + By + Cz = D
and
Ax + By + Cz = E
then the two planes are parallel.
(If we find that D = E, then the two planes are the same.)
Parallel planes
Most of the time, the planes will not be parallel, and will slope in different directions.
We want to find where these planes intersect.
That is, the point, or possible collection of points, which are on all of the planes.
Exercises
Exercises
Identify which of the following pairs of planes are parallel. Some algebraic manipulation may be required.
A: x + y + z = 2
B: 6x + 4y - 2z = 3
C: 3x + 2y - z = 10
D: x + 2y + 0.4z = 12
E: y - z = 2
F: z = 4 - x - y
G: z = 4x - 3y - 5
H: 150x = 24z
I: x - 0.75y - 0.25z = 1.25
J: 2.5x + 5y + z = 1
K: 2.5x - 0.4z = 7
L: y = z
Google Sites
Report abuse