https://file.aiquickdraw.com/mathbot/file/1726396228977nbm353ym.jpg

The image shows a parametric equation of a plane:

\[
\mathbf{r} = (2, 1, 4) + s(1, -2, 1) + t(1, -4, 3)
\]

You are tasked with converting this parametric form to the scalar (Cartesian) form of the plane's equation. You've already identified the point \(\mathbf{r_0} = (2, 1, 4)\) and the normal vector \(\mathbf{n} = (-2, -2, -2)\).

To convert the parametric form to scalar form, the equation of the plane can be written as:

\[
\mathbf{n} \cdot (\mathbf{r} - \mathbf{r_0}) = 0
\]

Given \(\mathbf{n} = (-2, -2, -2)\) and \(\mathbf{r_0} = (2, 1, 4)\), we substitute into the general form of the plane equation:

\[
-2(x - 2) - 2(y - 1) - 2(z - 4) = 0
\]

Simplifying:

\[
-2x + 4 - 2y + 2 - 2z + 8 = 0
\]
\[
-2x - 2y - 2z + 14 = 0
\]
Dividing the entire equation by \(-2\):

\[
x + y + z - 7 = 0
\]

Thus, the scalar form of the equation is:

\[
x + y + z = 7
\]

Would you like further details or have any questions? Here are some related questions to consider:

1. How is the normal vector to a plane determined from the parametric form?
2. Why does the dot product of the normal vector and a point on the plane yield zero?
3. What is the geometric meaning of the point-normal form of a plane?
4. How can parametric equations be used in vector calculus?
5. How does this method generalize to finding the equation of a plane in 3D space?

**Tip:** When converting parametric forms to scalar forms, always check that the vector used as the normal is perpendicular to both direction vectors in the parametric equation.

Consider the following parametric representation of a plane: r = (2, 1, 4) + s(1, -2, 1) + t(1, -4, 3). Enter vectors r_0 and n such that (r - r_0) • n = 0 is the point-normal form of the plane r. Now enter the scalar (Cartesian) form of the equation.

Learn how to convert a parametric plane equation r = (2, 1, 4) + s(1, -2, 1) + t(1, -4, 3) to the scalar (Cartesian) form of a plane. Using the point r_0 and normal vector n, we derive the equation x + y + z = 7, a fundamental concept in vector calculus and 3D geometry. Suitable for Grades 11-12.

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