If the planes b x-a y=n, c y-b z=l and a z-c x=m intersect in a line, then a l+b m+c n is equal

step 1 I have given here the planes here we have these three planes here they intersect in a line. So t

If the planes b x-a y=n, c y-b z=l and a z-c x=m intersect...

Key Concepts

Invariant Expressions from Geometric Conditions

In problems involving the intersection of geometrical objects, certain combinations or linear combinations of the coefficients and constants from the equations remain invariant under the given condition of intersection. Recognizing such invariant expressions helps in deducing properties that are independent of the particular coordinates selected, and thus are central in solving geometrical problems by reducing the complexity to checking these conditions.

Consistency Conditions in Linear Systems

When planes are represented by linear equations, finding their intersection is equivalent to solving a system of linear equations. The existence of a line of intersection indicates that the system is underdetermined and has a one-parameter family of solutions. This situation arises from specific consistency conditions among the coefficients and constants, often revealed through the examination of determinants or dependency relationships in the system.

Intersection of Planes in Three Dimensions

This concept deals with how multiple planes can intersect in space. The intersection of three planes may result in a unique point, a line, or even an entire plane, depending on the relationships between their equations. In the case where the planes intersect in a line, there is a degree of dependency among the equations that reduces the system’s rank, leading to infinitely many solutions that form a straight line.

Plane Equations and Normal Vectors

A plane in three-dimensional space is represented by a linear equation whose coefficients correspond to the components of a normal vector perpendicular to the plane. These normal vectors are crucial in determining the orientation of the plane and the nature of its intersections with other planes. Their relative directions and the conditions of coplanarity or linear dependence play a key role in understanding intersections such as lines or points.

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