Describing a plane in three dimensions

Question

When describing a plane in three dimensions, one uses a point P and a vector N normal to the plane, where N describes the "tilt" or orientation of the plane.Is it possible to describe a plane using a point P and a vector V parallel to the plane? Wouldn't V just be a translation of some vector W on the plane? If you can't use a vector parallel to the plane, why can't you?

Answer 1

What you need is a point and two direction vectors:

$$\mathbf r(s,t) = \mathbf p + s\mathbf u + t\mathbf v$$

Here $\mathbf p$ is the vector pointing to the point $P$ and $\mathbf u$ and $\mathbf v$ are the two noncollinear, nonzero vectors parallel to the plane.

The reason you need two direction vectors is that a plane is two-dimensional.

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Compare this to the parametrization of a line:

$$\mathbf r(t) = \mathbf p + t\mathbf v$$

Can you see how to parametrize any $n$-dimensional flat in a similar way?

Answer 2

No, you cannot: in $\Bbb R^3$ there are infinitely many planes through a given point $P$ and parallel to a given nonzero vector $V$. They are parameterized, e.g., by the set of directions orthogonal to $V$.