find parametric equation for the line through P0=(3,-1,1) perpendicular to the plane 3x+5y-7z=29

To find the parametric equations of a line through a point and perpendicular to a plane, follow these steps:

### Step 1: Identify the normal vector of the plane.
The equation of the plane is \( 3x + 5y - 7z = 29 \). The normal vector \(\mathbf{n}\) to the plane is given by the coefficients of \(x\), \(y\), and \(z\) in the plane equation. Thus, the normal vector is:
\[
\mathbf{n} = \langle 3, 5, -7 \rangle
\]
This normal vector is also the direction vector for the line, because the line is perpendicular to the plane.

### Step 2: Use the point \( P_0(3, -1, 1) \).
The point through which the line passes is \( P_0 = (3, -1, 1) \).

### Step 3: Write the parametric equations of the line.
The parametric equations for a line are given by:
\[
x = x_0 + at
\]
\[
y = y_0 + bt
\]
\[
z = z_0 + ct
\]
where \( (x_0, y_0, z_0) \) is a point on the line, \(\langle a, b, c \rangle\) is the direction vector, and \( t \) is the parameter.

In this case:
- Point \( P_0 = (3, -1, 1) \),
- Direction vector \( \mathbf{n} = \langle 3, 5, -7 \rangle \).

So, the parametric equations are:
\[
x = 3 + 3t
\]
\[
y = -1 + 5t
\]
\[
z = 1 - 7t
\]

### Final Parametric Equations:
\[
x = 3 + 3t
\]
\[
y = -1 + 5t
\]
\[
z = 1 - 7t
\]
where \( t \in \mathbb{R} \).

Would you like a more detailed explanation of any step?

### 5 Related Questions:
1. How do you find the direction vector from a plane equation?
2. What is the geometric interpretation of the normal vector to a plane?
3. How can you find the parametric equation of a line given two points?
4. What happens when a line is parallel to a plane instead of perpendicular?
5. How can you determine if a point lies on a given line?

### Tip:
The direction vector of a line perpendicular to a plane is always the normal vector of that plane.

Find the parametric equation for the line through P0=(3,-1,1) perpendicular to the plane 3x+5y-7z=29.

Find the parametric equation for a line through the point P0=(3,-1,1) that is perpendicular to the plane 3x + 5y - 7z = 29. Learn about normal vectors, parametric equations, and vector geometry. Suitable for students in Grades 10-12.

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