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find equation of plane passing through point perpendicular to given vector/line

To find the equation of a plane passing through the point \((-1, 4, 0)\) and perpendicular to the given vector/line, we can follow these steps:

1. **Extract the direction vector from the parametric equations:**
   The parametric equations given are:
   \[
   x = -1 + 2t, \quad y = 5 - t, \quad z = 3 - 2t
   \]
   The coefficients of \(t\) in these equations represent the direction vector of the line. Therefore, the direction vector \(\mathbf{v}\) is:
   \[
   \mathbf{v} = \langle 2, -1, -2 \rangle
   \]

2. **Use the direction vector as the normal vector to the plane:**
   Since the plane is perpendicular to the given line, the direction vector of the line is the normal vector \(\mathbf{n}\) to the plane. Hence, \(\mathbf{n} = \langle 2, -1, -2 \rangle\).

3. **Form the equation of the plane:**
   The general equation of a plane in the form \(Ax + By + Cz = D\) can be written using the normal vector \(\mathbf{n} = \langle A, B, C \rangle\) and a point \((x_0, y_0, z_0)\) that lies on the plane.

   For the point \((-1, 4, 0)\) and the normal vector \(\langle 2, -1, -2 \rangle\), the equation of the plane is:
   \[
   2(x + 1) - 1(y - 4) - 2(z - 0) = 0
   \]

4. **Simplify the equation:**
   Expand and simplify the equation:
   \[
   2(x + 1) - 1(y - 4) - 2z = 0
   \]
   \[
   2x + 2 - y + 4 - 2z = 0
   \]
   \[
   2x - y - 2z + 6 = 0
   \]

Thus, the equation of the plane passing through the point \((-1, 4, 0)\) and perpendicular to the given line is:
\[
2x - y - 2z + 6 = 0
\]

Would you like more details or have any questions about this solution? 

Here are some related questions that you might find interesting:
1. How do you find the equation of a line passing through two points in 3D space?
2. What is the geometric interpretation of the dot product in relation to vectors?
3. How do you determine if two planes are parallel or perpendicular?
4. What is the significance of the cross product of two vectors in 3D space?
5. How do you find the distance from a point to a plane in 3D space?
6. Can you explain how to find the intersection line of two planes?
7. How do you convert parametric equations of a line to its symmetric form?
8. What is the normal form of the equation of a plane and how is it derived?

**Tip:** When dealing with equations of planes, always check if your normal vector is correctly oriented by verifying it against given constraints (like perpendicularity or parallelism).

Learn how to find the equation of a plane passing through a point perpendicular to a given vector or line in 3D space. This problem involves using vectors, parametric equations, and plane geometry concepts. Suitable for advanced high school students and those studying 3D geometry.

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