obtain the Cartesian equation of the plane that passes through the points (3, 5, 1), (2, 2, 2)
and (−2, −2, 0).

To find the Cartesian equation of the plane passing through the points \((3, 5, 1)\), \((2, 2, 2)\), and \((-2, -2, 0)\), follow these steps:

---

### **Step 1: Find two direction vectors in the plane**
Given points, we form two vectors that lie in the plane:
\[
\vec{v_1} = (2 - 3, 2 - 5, 2 - 1) = (-1, -3, 1)
\]
\[
\vec{v_2} = (-2 - 3, -2 - 5, 0 - 1) = (-5, -7, -1)
\]

---

### **Step 2: Compute the normal vector**
The normal vector to the plane, \(\vec{n}\), is perpendicular to both \(\vec{v_1}\) and \(\vec{v_2}\). Compute it using the cross product:
\[
\vec{n} = \vec{v_1} \times \vec{v_2}
\]

The determinant for the cross product is:
\[
\vec{n} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-1 & -3 & 1 \\
-5 & -7 & -1
\end{vmatrix}
\]

Expanding:
\[
\vec{n} = \mathbf{i} \begin{vmatrix} -3 & 1 \\ -7 & -1 \end{vmatrix} 
- \mathbf{j} \begin{vmatrix} -1 & 1 \\ -5 & -1 \end{vmatrix} 
+ \mathbf{k} \begin{vmatrix} -1 & -3 \\ -5 & -7 \end{vmatrix}
\]

Simplify the minors:
\[
\vec{n} = \mathbf{i}((-3)(-1) - (1)(-7)) 
- \mathbf{j}((-1)(-1) - (-5)(1)) 
+ \mathbf{k}((-1)(-7) - (-3)(-5))
\]

\[
\vec{n} = \mathbf{i}(3 + 7) - \mathbf{j}(1 + 5) + \mathbf{k}(-7 - 15)
\]
\[
\vec{n} = 10\mathbf{i} - 6\mathbf{j} - 22\mathbf{k}
\]

Thus, the normal vector is:
\[
\vec{n} = (10, -6, -22)
\]

---

### **Step 3: Form the Cartesian equation of the plane**
The general equation of a plane is:
\[
10x - 6y - 22z = d
\]
Substitute the coordinates of a point (e.g., \((3, 5, 1)\)) into the equation to find \(d\):
\[
10(3) - 6(5) - 22(1) = d
\]
\[
30 - 30 - 22 = d
\]
\[
d = -22
\]

Thus, the equation of the plane is:
\[
10x - 6y - 22z = -22
\]

---

### **Final Answer**
The Cartesian equation of the plane is:
\[
10x - 6y - 22z + 22 = 0
\]

Would you like further explanation or assistance with any step? 😊  

---

### **Related Questions**
1. How do you compute the cross product of two vectors geometrically?
2. What does the normal vector of a plane represent in 3D space?
3. How can the distance from a point to a plane be calculated?
4. What are the parametric equations of the same plane?
5. How can you verify if a given point lies on a plane?

---

### **Tip**
When computing the cross product, carefully organize the determinant's expansion to avoid sign errors.

Obtain the Cartesian equation of the plane that passes through the points (3, 5, 1), (2, 2, 2), and (−2, −2, 0).

Learn how to find the Cartesian equation of the plane passing through the points (3, 5, 1), (2, 2, 2), and (−2, −2, 0). This solution involves vector algebra concepts like direction vectors and cross products, suitable for 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