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To find the equation of a plane through three points, \((5, -8, -1)\), \((-8, 4, 5)\), and \((-6, -3, 6)\), we will follow these steps:

### 1. Use the general equation of a plane:
\[
Ax + By + Cz + D = 0
\]
where \((A, B, C)\) is the normal vector to the plane.

### 2. Compute two vectors in the plane:
Let's calculate two vectors from the given points:
- \(\vec{v_1} = (-8 - 5, 4 - (-8), 5 - (-1)) = (-13, 12, 6)\)
- \(\vec{v_2} = (-6 - 5, -3 - (-8), 6 - (-1)) = (-11, 5, 7)\)

### 3. Find the normal vector \(\vec{n} = \vec{v_1} \times \vec{v_2}\):
The cross product \(\vec{v_1} \times \vec{v_2}\) gives us the normal vector \(\vec{n} = (A, B, C)\).

I will calculate the cross product to find the normal vector.The normal vector \(\vec{n} = (54, 25, 67)\). So the equation of the plane is:

\[
54x + 25y + 67z + D = 0
\]

### 4. Find \(D\):
We substitute the coordinates of one of the given points, say \((5, -8, -1)\), into the plane equation to solve for \(D\):

\[
54(5) + 25(-8) + 67(-1) + D = 0
\]

Let's calculate \(D\).The value of \(D\) is \(-3\).

### Final equation of the plane:
\[
54x + 25y + 67z - 3 = 0
\]

This is the equation of the plane passing through the points \((5, -8, -1)\), \((-8, 4, 5)\), and \((-6, -3, 6)\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the angle between two planes given their normal vectors?
2. How can you determine if a point lies on a given plane?
3. What is the geometric significance of the normal vector to a plane?
4. How do you find the distance from a point to a plane?
5. How can we check if two planes are parallel?

**Tip:** The cross product of two vectors gives a vector that is perpendicular to both, which is useful in finding normals to planes.

Find the equation of a plane through the points (5, -8, -1), (-8, 4, 5), and (-6, -3, 6).

Learn how to find the equation of a plane passing through the points (5, -8, -1), (-8, 4, 5), and (-6, -3, 6) by using vector cross products and solving for the normal vector. Suitable for students in Grades 10-12, this step-by-step guide covers linear algebra concepts and vector calculations.

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