In Problems, find the point of intersection of the given plane and line. x+y+4 z=12 ; x=3-2 t, y

step 1 So in this question in problem find point of intersection of given plane and line x plus y plus

In Problems, find the point of intersection of the given...

Key Concepts

Plane Equation

A plane in three-dimensional space can be defined by a linear equation in x, y, and z, usually expressed in the form ax + by + cz = d. This formulation represents a flat, two-dimensional surface and is characterized by its normal vector, which is perpendicular to the plane. The constants in the equation determine the orientation and position of the plane in space.

Parametric Equation of a Line

A line in three-dimensional space is often described using parametric equations, where each coordinate (x, y, and z) is expressed as a function of a single parameter, typically denoted as t. This representation specifies a starting point on the line and a direction vector that indicates the line's orientation. It provides a convenient way to generate all points along the line by varying the parameter.

Intersection of a Line and a Plane

Finding the intersection between a line and a plane involves determining the point that lies on both geometric objects. This is done by substituting the parametric expressions of the line into the plane's equation, which yields an equation involving the parameter. Solving this equation gives the specific value of the parameter at which the line meets the plane, and substituting that value back into the line's parametric equations results in the coordinates of the intersection point.

Substitution Method

The substitution method is an algebraic technique used to solve systems of equations by replacing one variable with an equivalent expression derived from another equation. In the context of finding the intersection of a line and a plane, the coordinates given by the line's parametric equations are substituted into the plane's equation. This leads to an equation in a single variable, making it easier to solve for the parameter that defines the intersection point.

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