The planes with the equations x+2 z=12 and y-3 z=6 intersect in a line. Find the equation for th

step 1 Okay, so in this problem, what we're going to have to do. So in a circle whose center is at T, t

The planes with the equations x+2 z=12 and y-3 z=6 intersect...

Key Concepts

Parametric Representation of a Line

A line in three dimensions can be expressed in parametric form as (x, y, z) = (x?, y?, z?) + t(a, b, c), where (x?, y?, z?) is a specific point on the line and (a, b, c) is the direction vector. This representation highlights both a point through which the line passes and the direction in which the line extends, making it a versatile format for describing lines in space.

Direction Vector

The direction vector of a line specifies the orientation and slope of the line in space. It is crucial in parameterizations because it determines how the line 'moves' from its base point. In cases where the line is the intersection of two planes, the direction vector can be found by solving the system of equations or taking the cross product of the normals of the two planes.

Intersection of Planes

The intersection of two planes in three-dimensional space occurs when the set of points satisfying both plane equations simultaneously forms a line. This concept is fundamental in understanding how different geometric objects relate and interact in space, particularly when solving for the common solution of multiple equations.

System of Equations

Solving the intersection of planes involves dealing with a system of linear equations. The goal is to find a consistent set of values for the variables that satisfy both equations simultaneously. Often, this process involves isolating a free variable and expressing the other variables in terms of it, which naturally leads to the parametric representation of the intersecting line.

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