Math Problem Statement
Write the equation of the plane with a normal vector n
⟨ 1 , − 5 , 1 ⟩ and the point (1, 2, 3) lying on the plane.
Solution
To write the equation of a plane, we use the general formula for the equation of a plane:
Where:
- is the normal vector to the plane.
- is a point on the plane.
Given that the normal vector and the point lies on the plane, we can substitute , , and into the plane equation. The point will help us find .
The equation becomes:
Now simplify it:
Expand the terms:
Combine like terms:
Thus, the equation of the plane is:
Would you like more details or have any questions?
Here are 5 related questions:
- How do you find the normal vector from a given plane equation?
- What is the equation of a plane parallel to another plane?
- How can you determine if two planes are perpendicular?
- What happens when a point is substituted into the plane equation?
- How do you find the distance between a point and a plane?
Tip: Always check that the normal vector is perpendicular to the plane for any given plane equation.
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Math Problem Analysis
Mathematical Concepts
Geometry
Planes
Vectors
Formulas
Equation of a plane: Ax + By + Cz = D
Theorems
Plane equation theorem based on a normal vector and a point on the plane
Suitable Grade Level
Grades 10-12