write an equation for a plane containing the points (1,-2,3), (5,1,1) and (3,-3,1)

To find the equation of a plane containing the points \( A(1, -2, 3) \), \( B(5, 1, 1) \), and \( C(3, -3, 1) \), we can follow these steps:

### Step 1: Find two vectors in the plane
We first need to find two vectors in the plane using the given points:
- Vector \( \vec{AB} \) from \( A \) to \( B \):
\[
\vec{AB} = B - A = (5 - 1, 1 - (-2), 1 - 3) = (4, 3, -2)
\]
- Vector \( \vec{AC} \) from \( A \) to \( C \):
\[
\vec{AC} = C - A = (3 - 1, -3 - (-2), 1 - 3) = (2, -1, -2)
\]

### Step 2: Find the normal vector to the plane
The normal vector \( \vec{n} \) to the plane is given by the cross product of vectors \( \vec{AB} \) and \( \vec{AC} \).

\[
\vec{n} = \vec{AB} \times \vec{AC} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & 3 & -2 \\
2 & -1 & -2 \\
\end{vmatrix}
\]
This is calculated as follows:

\[
\vec{n} = \mathbf{i} \left(3 \times -2 - (-1 \times -2)\right) - \mathbf{j} \left(4 \times -2 - 2 \times -2\right) + \mathbf{k} \left(4 \times -1 - 3 \times 2\right)
\]
\[
\vec{n} = \mathbf{i}(-6 - 2) - \mathbf{j}(-8 + 4) + \mathbf{k}(-4 - 6)
\]
\[
\vec{n} = \mathbf{i}(-8) - \mathbf{j}(-4) + \mathbf{k}(-10)
\]
\[
\vec{n} = (-8, 4, -10)
\]

So the normal vector \( \vec{n} \) to the plane is \( (-8, 4, -10) \).

### Step 3: Write the equation of the plane
The equation of a plane can be written in the form:
\[
Ax + By + Cz = D
\]
Where \( (A, B, C) \) is the normal vector and \( (x_0, y_0, z_0) \) is a point on the plane.

Using point \( A(1, -2, 3) \) and the normal vector \( (-8, 4, -10) \), we substitute into the equation:
\[
-8(x - 1) + 4(y + 2) - 10(z - 3) = 0
\]
Simplifying:
\[
-8x + 8 + 4y + 8 - 10z + 30 = 0
\]
\[
-8x + 4y - 10z + 46 = 0
\]
Thus, the equation of the plane is:
\[
8x - 4y + 10z = 46
\]

Let me know if you need further clarification!

### Related Questions:
1. How do you compute the cross product of two vectors?
2. What is the geometric significance of the normal vector to a plane?
3. Can a plane in 3D space be represented using parametric equations?
4. How do you verify if a point lies on a given plane?
5. What are the different forms of representing the equation of a plane?

### Tip:
When solving for the equation of a plane, the normal vector is key since it determines the plane's orientation in space.

Learn how to derive the equation of a plane passing through three points: (1, -2, 3), (5, 1, 1), and (3, -3, 1). This tutorial covers key concepts in vector geometry, including how to compute the cross product and use it to find the normal vector of a plane. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation