21-32= Find an equation of the plane. The plane that passes through the point (-1,2,1) and conta

step 1 We need to find the equation of plan such that it contains a point minus 1 to 1 and it contains

21-32= Find an equation of the plane. The plane that passes...

Key Concepts

Plane Equation in 3D Space

A plane in three-dimensional space can be defined using a point and a normal vector. Its equation in scalar form is often written as Ax + By + Cz + D = 0, where (A, B, C) is the normal vector perpendicular to the plane. This representation is fundamental in many areas of geometry and linear algebra as it categorizes the orientation and position of the plane in space.

Line of Intersection of Two Planes

When two non-parallel planes intersect, they do so along a line. The line of intersection satisfies the equations of both planes simultaneously. Determining this line involves solving the system of equations defined by the two planes, which is a significant concept in understanding how multiple planes relate in three dimensions.

Family of Planes Containing a Common Line

A family of planes can be constructed that all share a common line, such as the intersection line of two given planes. This is normally done by taking a linear combination of the original plane equations. The idea is that any plane in the family will automatically contain the intersection line, and additional conditions (like passing through a specific point) can be used to pinpoint a unique plane within that family.

Point Substitution for Determining Specific Planes

Once a general equation for a family of planes is obtained (typically containing a free parameter), a specific plane in that family can be chosen by substituting the coordinates of a known point into the equation. This substitution provides a condition that allows for the determination of the free parameter, yielding the unique equation of the desired plane.

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