Determine whether the line and plane intersect; if so, find...

Determine whether the line and plane intersect; if so, find the coordinates of the intersection.
$$
\begin{array}{l}{\text { (a) } x=t, y=t, z=t} \\ {3 x-2 y+z-5=0} \\ {\text { (b) } x=2-t, y=3+t, z=t} \\ {\quad 2 x+y+z=1}\end{array}
$$

Key Concepts

Parametric Representation of a Line

This concept involves describing a line in space using one or more parameters. Each coordinate (x, y, z) is expressed as a function of a common parameter (usually t), which allows one to capture the direction and position of the line in three-dimensional space. It is crucial in determining specific points on the line and for substituting into other geometric equations.

Cartesian Equation of a Plane

A plane in three-dimensional space can be represented by a linear equation in the form Ax + By + Cz + D = 0. This equation defines all points (x, y, z) that lie on the plane. Understanding this representation is essential for analyzing the spatial relationship between the plane and other geometric objects like lines.

Solving Systems of Equations

When determining the intersection of a line and a plane, one typically substitutes the parametric equations of the line into the plane's equation. This process creates a system of equations, usually resulting in a single equation in one unknown (the parameter). Solving this equation reveals the parameter value at which the line meets the plane, if an intersection exists.

Intersection of a Line and Plane

Finding the intersection involves combining the parametric form of the line with the plane's equation through substitution. If the resulting equation has a solution, it yields a specific value of the parameter that, when substituted back into the parametric equations, gives the coordinates of the intersection point. This process is fundamental for analyzing the relative positioning of geometric entities in space.

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