Find an equation of the plane. The plane that passes through the point (-1,2,1) and contains the

step 1 Okay, doing with the next question, we have the equation of planes x plus y minus z equals to 2.

Find an equation of the plane. The plane that passes through...

Key Concepts

Linear Combination of Plane Equations

A linear combination of plane equations involves adding or subtracting scaled versions of the equations of two planes. This strategy is used to find an equation for a new plane that not only includes the line of intersection of the original planes but also satisfies additional conditions, such as passing through a specific point. It leverages the fact that any plane containing the intersection can be expressed as a combination of the given plane equations.

Line of Intersection of Planes

The line of intersection is formed when two non-parallel planes meet. This line lies in both planes and can be described by a direction vector and a point that belongs to both planes. Understanding the line of intersection is key when finding a plane that contains this line, as any plane including the line must also include its directional vector.

Plane Equation

A plane equation is a mathematical representation that defines a flat, two-dimensional surface in three-dimensional space. It is typically expressed in the form Ax + By + Cz = D, where [A, B, C] is the normal vector perpendicular to the plane and D is a constant. This form is useful for determining if points lie on the plane and for analyzing the orientation of the plane in space.

Normal Vector

The normal vector is a vector that is perpendicular to a plane and plays a crucial role in the plane’s equation. It provides directional information about the orientation of the plane, and two planes are parallel if their normal vectors are proportional. Knowing the normal vector allows one to write the plane's equation using the point-normal form, which is particularly useful when a specific point on the plane is known.

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