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

step 1 The line equation in parameter form is given as x is equal to 1 plus t is a parameter y is equal

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

Key Concepts

Parametric Representation of a Line

This concept involves expressing the points on a line in three-dimensional space using a parameter, usually denoted as t. The parametric form gives the coordinates as functions of t, which highlights the line's starting point and direction vector.

Direction Vector

The direction vector is derived from the coefficients of the parameter in the line's parametric equations. It indicates the direction in which the line extends and is essential for determining the orientation of planes that contain the line.

Plane Equation in Standard Form

A plane can be represented by an equation of the form Ax + By + Cz = D, where (A, B, C) is a normal vector perpendicular to the plane. This standard form is widely used because it clearly exhibits the geometric properties of the plane.

Normal Vector

The normal vector of a plane is a vector that is perpendicular to every line lying in that plane. It is crucial for defining the plane's orientation, and it also helps in determining if a given line lies within the plane or the intersection of two planes.

Intersection of Planes

When two distinct planes intersect, they do so along a line. Understanding the conditions that allow two planes to intersect along a given line involves ensuring that both plane equations are consistent with the parametric equations of that line while having different normal vectors.

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