The planes with the equations x+3 z=12 and y-5 z=10 intersect in a line. Find the equation for t

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The planes with the equations x+3 z=12 and y-5 z=10...

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Key Concepts

Intersection of Planes

The intersection of two planes in three-dimensional space is a line (provided the planes are not parallel or coincident). Determining this intersection involves finding the set of points that simultaneously satisfies both plane equations, essentially solving a system of linear equations in three variables.

Parametric Equation of a Line

A line in space can be expressed in parametric form, where each coordinate (x, y, z) is written as the sum of a fixed point (a point on the line) and a scalar multiple of a direction vector. This form highlights the fact that moving along the line involves scaling the direction vector from the given point.

Normal Vector

Each plane in three-dimensional space is defined by a normal vector, which is perpendicular to the plane. The normal vector is derived directly from the coefficients of the plane's equation. It plays a crucial role in geometrical analysis, particularly when determining angles between planes or finding the direction of the line of intersection.

Cross Product

The cross product of two vectors results in a vector that is orthogonal to both. When applied to the normal vectors of two planes, the cross product yields a direction vector for the line of intersection, capturing the directional alignment of the line where the two planes meet.

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