Find two different planes whose intersection is the line x=1+t, y=2-t, z=3+2 t . Write equations

step 1 given the equation of the line so the line here x equal to 1 plus t y equal to 2 minus t and z e

Find two different planes whose intersection is the line...

Key Concepts

Normal Vectors

A normal vector is a vector that is perpendicular to a given plane. It plays a crucial role in defining the orientation of a plane and forms the basis for writing the plane's equation. The components of the normal vector directly correspond to the coefficients in the plane’s linear equation, encapsulating the plane's directional properties.

Intersection of Two Planes

When two non-parallel planes in three-dimensional space intersect, their intersection is a line. This concept explores how the geometric properties of the planes, particularly their normal vectors and positions, result in a one-dimensional intersection where every point satisfies both plane equations.

Parametric Representation of a Line

This concept involves expressing the coordinates of a line as functions of a parameter, typically denoted as t. It defines a point on the line and a direction vector, allowing for a concise description of all points along the line in three-dimensional space.

Plane Equation in Three-Dimensional Space

A plane in three-dimensional space is commonly described by a linear equation of the form Ax + By + Cz = D. This equation utilizes a normal vector—whose components are A, B, and C—which is perpendicular to the plane, and a scalar D that adjusts the plane's position relative to the origin.

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