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The problem asks to graph the points \( A(1,2,0) \), \( B(3,1,4) \), and \( C(2,3,5) \) in \( \mathbb{R}^3 \) and find the equation of the plane that passes through these three points.

### Step-by-Step Solution:

#### 1. **Determine the Vectors:**
To find the equation of the plane, we first need two vectors that lie on the plane. These vectors can be obtained from the given points:

\[
\overrightarrow{AB} = B - A = (3-1, 1-2, 4-0) = (2, -1, 4)
\]
\[
\overrightarrow{AC} = C - A = (2-1, 3-2, 5-0) = (1, 1, 5)
\]

#### 2. **Find the Normal Vector:**
The normal vector \( \mathbf{n} \) to the plane is found by taking the cross product of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \):

\[
\mathbf{n} = \overrightarrow{AB} \times \overrightarrow{AC} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & -1 & 4 \\
1 & 1 & 5
\end{vmatrix}
\]

Now, calculating the determinant:

\[
\mathbf{n} = \mathbf{i} \left( (-1)(5) - (1)(4) \right) - \mathbf{j} \left( (2)(5) - (1)(4) \right) + \mathbf{k} \left( (2)(1) - (-1)(1) \right)
\]
\[
\mathbf{n} = \mathbf{i}(-5 - 4) - \mathbf{j}(10 - 4) + \mathbf{k}(2 + 1)
\]
\[
\mathbf{n} = \mathbf{i}(-9) - \mathbf{j}(6) + \mathbf{k}(3)
\]
\[
\mathbf{n} = (-9, -6, 3)
\]

So, the normal vector to the plane is \( \mathbf{n} = (-9, -6, 3) \).

#### 3. **Equation of the Plane:**
The general equation of a plane is:

\[
Ax + By + Cz + D = 0
\]

Where \( (A, B, C) \) is the normal vector to the plane. Substituting \( \mathbf{n} = (-9, -6, 3) \), we have:

\[
-9x - 6y + 3z + D = 0
\]

To find \( D \), substitute the coordinates of one of the points, say \( A(1,2,0) \):

\[
-9(1) - 6(2) + 3(0) + D = 0
\]
\[
-9 - 12 + D = 0
\]
\[
D = 21
\]

Thus, the equation of the plane is:

\[
-9x - 6y + 3z + 21 = 0
\]

Or simplified:

\[
3x + 2y - z = 7
\]

### Final Answer:
The equation of the plane that passes through the points \( A(1,2,0) \), \( B(3,1,4) \), and \( C(2,3,5) \) is:

\[
3x + 2y - z = 7
\]

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Would you like further details or have any questions?

Here are 5 related questions:
1. How do you compute a cross product in general?
2. What are the geometric interpretations of the normal vector?
3. Can a plane be uniquely determined by just two points?
4. How can we verify if a point lies on a plane?
5. What are other methods to find the equation of a plane?

**Tip:** Always verify your plane equation by substituting all given points into the final equation.

Un ingeniero desea resaltar algunos puntos a lo largo de un plano tridimensional. Sabiendo que los puntos a resaltar son A(1,2,0), B(3,1,4), C(2,3,5). Graficar estos puntos en R^3 y encontrar la ecuación del plano que pasa por los tres puntos.

Learn how to graph points A(1,2,0), B(3,1,4), and C(2,3,5) in R^3 and find the equation of the plane that passes through them. This solution involves vectors, cross products, and finding the equation of a plane. Suitable for college-level students studying linear algebra.

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