91Ó°ÊÓ

Parametric equations are a way to describe a line using a set of equations, each dependent on a single parameter, often denoted as \( t \). This method is particularly useful when dealing with lines in three-dimensional space. The equations express the coordinates \( x \), \( y \), and \( z \) of any point on the line in terms of \( t \). For example, the parametric equations for a line might be given as:

• \( x = 3t \)
• \( y = 5t \)
• \( z = -t \)

These equations tell us that as the parameter \( t \) changes, the point \(( x, y, z )\) moves along the line. In essence, by substituting different values of \( t \), you can find any point on the line.
In our exercise, solving for \( t \) helps us determine if, and where, the line intersects with a given plane. By substituting the expressions for \( x \), \( y \), and \( z \) from the parametric forms into the plane equation, we can check whether a common solution exists, indicating an intersection point.

A plane equation in three-dimensional space can generally be written in the form \( ax + by + cz + d = 0 \). Here, \( a \), \( b \), \( c \), and \( d \) are constants, and \( x \), \( y \), and \( z \) are coordinates of any point on the plane. This equation represents a flat, two-dimensional surface extending infinitely in three dimensions.
The task with a problem involving a line and a plane is to determine if there is an intersection, which would represent a point satisfying both the plane and the line's equations.
In the given example, consider the plane equation \( 2x - y + z + 1 = 0 \). By substituting values from the parametric line equations, we are essentially finding whether there is a specific \( t \) that results in a true statement. If such a \( t \) exists, the corresponding \( x \), \( y \), and \( z \) will be coordinates of the intersection point. If the substitution results in a contradiction, as in the first part of the original exercise, it means no intersection occurs.

When solving for the intersection of a line and a plane, it essentially translates into solving a system of equations. The goal is to find a common solution, a set of \( x, y, \) and \( z \) values, that satisfies both the parametric equations of the line and the equation of the plane.

• Using substitution: Insert the expressions for \( x \), \( y \), and \( z \) from the line's parametric equations into the plane equation.
• Solve for \( t \): Simplify the resulting equation to find the corresponding \( t \) value that holds the equation true. This \( t \) gives us the point of intersection.

This approach results in a system of equations due to the dependency on \( t \). Once \( t \) is found, substituting it back into the parametric equations gives us the actual intersection point. In cases where no solution exists, like an inconsistency or contradiction, it shows that the line is parallel to the plane and does not intersect. Conversely, a valid solution exists where the line actually penetrates the plane, providing definite coordinates for the intersection.