State a formula for the parametrization of the plane that goes through the
point p and contains the vectors v and w.

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.

State a formula for the parametrization of the plane that goes through the point p and contains the vectors v and w.

Discover how to parametrize a plane that passes through a point p and contains two vectors v and w. This solution explains the formula r(s, t) = p + s*v + t*w with clear examples, ideal for multivariable calculus students.

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