n implicit equation for the plane passing through the points (5,2,−2), (0,7,−5), and (4,1,−3) is .

To find the implicit equation of a plane passing through three points, we use the following general equation of a plane:

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

### Step-by-step process:

1. **Label the given points:**
   \[
   P_1 = (5, 2, -2), \quad P_2 = (0, 7, -5), \quad P_3 = (4, 1, -3)
   \]

2. **Find two vectors that lie on the plane** by subtracting coordinates of the points:
   \[
   \vec{v_1} = P_2 - P_1 = (0-5, 7-2, -5 - (-2)) = (-5, 5, -3)
   \]
   \[
   \vec{v_2} = P_3 - P_1 = (4-5, 1-2, -3 - (-2)) = (-1, -1, -1)
   \]

3. **Find the normal vector** to the plane by taking the cross product of \(\vec{v_1}\) and \(\vec{v_2}\):
   \[
   \vec{n} = \vec{v_1} \times \vec{v_2}
   \]
   The cross product \(\vec{v_1} \times \vec{v_2}\) is computed as:
   \[
   \vec{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -5 & 5 & -3 \\ -1 & -1 & -1 \end{vmatrix}
   \]
   Expanding this determinant:
   \[
   \vec{n} = \mathbf{i} \left( 5(-1) - (-3)(-1) \right) - \mathbf{j} \left( -5(-1) - (-3)(-1) \right) + \mathbf{k} \left( -5(-1) - 5(-1) \right)
   \]
   \[
   \vec{n} = \mathbf{i}(-5 - 3) - \mathbf{j}(5 - 3) + \mathbf{k}(5 - 5)
   \]
   \[
   \vec{n} = \mathbf{i}(-8) - \mathbf{j}(2) + \mathbf{k}(0)
   \]
   \[
   \vec{n} = (-8, -2, 0)
   \]
   So, the normal vector \(\vec{n} = (-8, -2, 0)\).

4. **Write the equation of the plane:**

   Using the general form of the plane equation \(Ax + By + Cz + D = 0\) and the normal vector \(\vec{n} = (-8, -2, 0)\), the equation becomes:
   \[
   -8(x - 5) - 2(y - 2) + 0(z + 2) = 0
   \]
   Simplifying:
   \[
   -8x + 40 - 2y + 4 = 0
   \]
   \[
   -8x - 2y + 44 = 0
   \]
   So, the implicit equation of the plane is:
   \[
   8x + 2y = 44
   \]

Let me know if you'd like more details or clarifications.

---

Here are 5 related questions:
1. How do you find the cross product of two vectors in general?
2. What is the geometric meaning of the normal vector to a plane?
3. Can you derive the parametric equation of the plane using the same points?
4. What are the conditions for three points to lie on the same plane?
5. How do you find the distance from a point to a plane using the plane's equation?

**Tip:** When calculating cross products, using determinants can help simplify the process.

The query submitted by the user: Find the implicit equation for the plane passing through the points (5,2,−2), (0,7,−5), and (4,1,−3).

Learn how to find the implicit equation of a plane passing through three points: (5, 2, -2), (0, 7, -5), and (4, 1, -3). This solution uses vector operations, including the cross product, and explains the steps to derive the equation in 3D space. Suitable for 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