We discuss how to find implicit and explicit formulas for planes.

*slope*as being synonymous with lines and derivatives. In single variable calculus, the derivative is the slope of the tangent line. However, this identification of “slope” as being the “key-player” is something of a new idea. Before calculus, people were often interested in finding normal lines to curves. While this might seem odd at first, consider this: A line is the set of all points normal to some vector. Check it out:

is an implicit formula for the line. This means that if points make the equation above true, then those points are on the line. As we will see, a nearly identical implicit formula defines a plane.

### Implicit planes

We would like to know the implicit formula for a plane. Remember an implicit function in is one of the form: Here the dot product saves the day. Recall that if is any vector, and , then the equation is solved by all vectors that are orthogonal to . We plotted several such vectors below:

gives the formula for a plane. Since is a solution, this plane must pass through the origin. If we want our plane to be located anywhere in space, we must know a point on the plane, call it . Putting this together, we can now see:

If you know

- a vector and
- a point (given by a vector)

then,

is an implicit equation for a plane passing through the point with normal vector .

Normal vectors not only allow us to define equations for planes but also they help us describe properties of planes.

**parallel**if their normal vectors are parallel. Two planes are said to be

**orthogonal**if their normal vectors are orthogonal.

### Parametric planes

Given **any** two nonzero vectors in , and , such that we can produce a parametric
formula for a plane by writing where is a vector whose “tip” is on the plane, and
and are in the plane.

The vector-valued formula for a plane is very similar to our formula for a line, where is a vector that points in the direction of the line, both represent linear relationships, and hence we use similar notation for both.

Now that we have **two** methods of graphing planes, let’s use both of the
representations at once!