Consider the line x=5-2 t, y=3+7 t, z=4 t and the plane a...

Consider the line $x=5-2 t, y=3+7 t, z=4 t$ and the plane $a x+b y+c z=d$. All the following questions have many possible answers. Find values of $a, b, c, d$ such that: (a) The plane is perpendicular to the line. (b) The plane is perpendicular to the line and through the point (5,3,0) (c) The line lies in the plane.

Consider the line $x=5-2 t, y=3+7 t, z=4 t$ and the plane $a x+b y+c z=d$. All the following questions have many possible answers. Find values of $a, b, c, d$ such that:
(a) The plane is perpendicular to the line.
(b) The plane is perpendicular to the line and through the point (5,3,0)
(c) The line lies in the plane.

Key Concepts

Parametric Representation of a Line

This concept involves expressing a line in space using a parameter, typically denoted by t. In a three-dimensional setting, the line is represented through equations for x, y, and z coordinates as functions of t. It provides a clear way to identify both a point through which the line passes and its direction vector.

General Equation of a Plane

A plane in three-dimensional space can be described by a linear equation of the form ax + by + cz = d. Here, the coefficients a, b, and c form the normal vector to the plane, and d is a constant that determines the plane's position relative to the origin.

Normal Vector of a Plane

The normal vector is a vector that is perpendicular to every line lying in the plane. It is directly extracted from the coefficients (a, b, c) in the plane equation. This concept is critical because the relationship between the normal vector and the direction vector of a line can determine whether the line is perpendicular or parallel to the plane.

Perpendicularity (Orthogonality) Condition

When a plane is perpendicular to a line, the direction vector of the line is orthogonal to the plane. This is captured mathematically by ensuring that the direction vector of the line is parallel to the normal vector of the plane, often checked using the dot product condition.

Line Containment in a Plane

For a line to lie entirely within a plane, every point on the line must satisfy the plane's equation. In practical terms, substituting the parametric equations of the line into the equation of the plane should yield an identity in the parameter, ensuring the equation holds for all values of the parameter.

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