O(0,0,0), A(1,3,1). OA has for cartesian equations y=3x and z=x. the line d has for cartesian equations: x=z-1 and y=2z+1. what are the cartesian equations of the plans P and F so that d belongs to P and OA belongs to F? F and P are parallel.

To solve this problem, let's analyze the given information step by step.

### Step 1: Vector and Cartesian Equations of OA
The vector **OA** is defined by the point \( O(0,0,0) \) and the point \( A(1,3,1) \). The direction vector of **OA** can be found as:
\[
\overrightarrow{OA} = \begin{pmatrix} 1-0 \\ 3-0 \\ 1-0 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}
\]
The line **OA** has the parametric equations:
\[
x = t, \quad y = 3t, \quad z = t
\]
The corresponding Cartesian equations (as given) are:
\[
y = 3x \quad \text{and} \quad z = x
\]

### Step 2: Cartesian Equations of the Line d
The line \( d \) is given with Cartesian equations:
\[
x = z - 1 \quad \text{and} \quad y = 2z + 1
\]
The parametric equations of line \( d \) can be written as:
\[
\begin{aligned}
x &= t - 1, \\
y &= 2t + 1, \\
z &= t,
\end{aligned}
\]
where \( t \) is the parameter.

The direction vector of the line \( d \) is:
\[
\overrightarrow{d} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}
\]

### Step 3: Equation of Plane \( F \) Containing Line **OA**
Plane \( F \) must contain line **OA** and therefore must have a normal vector perpendicular to the direction vector of **OA**.

The direction vector of **OA** is:
\[
\overrightarrow{OA} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}
\]
A normal vector \( \overrightarrow{n_F} \) for plane \( F \) can be any vector orthogonal to \( \overrightarrow{OA} \). 

Since the Cartesian equations of a plane can be written as:
\[
ax + by + cz = d,
\]
we need to ensure \( \overrightarrow{n_F} \cdot \overrightarrow{OA} = 0 \).

Let's take \( \overrightarrow{n_F} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \), which is perpendicular to \( \overrightarrow{OA} \):
\[
\overrightarrow{n_F} \cdot \overrightarrow{OA} = 1(1) + (-1)(3) + 2(1) = 1 - 3 + 2 = 0
\]
The equation of plane \( F \) passing through \( O(0,0,0) \) is:
\[
x - y + 2z = 0
\]

### Step 4: Equation of Plane \( P \) Containing Line d
Plane \( P \) must contain line \( d \) and be parallel to plane \( F \). Therefore, \( P \

Explore how to determine the Cartesian equations of planes P and F in 3D space, given specific conditions involving vectors, parametric equations, and Cartesian equations of lines. Understand the parallelism condition and derive equations step-by-step. Suitable for advanced high school students.

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