The formula for the parametrization of a plane that passes through a point p and contains the vectors v and w is given by:

$\mathbf{r}(s, t) = \mathbf{p} + s \mathbf{v} + t \mathbf{w}$

Where:

• $\mathbf{r}(s, t)$ is the position vector of any point on the plane.
• $\mathbf{p}$ is the position vector of the given point $p$.
• $\mathbf{v}$ and $\mathbf{w}$ are two non-parallel vectors that lie on the plane.
• $s$ and $t$ are real parameters that vary over all real numbers to describe every point on the plane.

This equation expresses the plane parametrically using two independent variables, $s$ and $t$.

Would you like further explanation or examples? Here are 5 related questions to deepen your understanding:

1. How do you find the parametric equations of a plane given three points?
2. Can you express a plane in scalar form using the parametrization formula?
3. How do you check if a point lies on a parametrized plane?
4. What is the relationship between the normal vector of a plane and the vectors v and w?
5. How can you convert a parametric plane equation into its standard (Cartesian) form?

Tip: Ensure that the vectors v and w are linearly independent; otherwise, they will not span a plane.