(a) Find an equation for the line perpendicular to the plane...

(a) Find an equation for the line perpendicular to the plane $2 x-3 y=z$ and through the point (1,3,7).
(b) Where does the line cut the plane?
(c) What is the distance between the point (1,3,7) and the plane?

Key Concepts

Normal Vector of a Plane

This concept refers to a vector that is perpendicular to a plane. It is often determined directly from the coefficients of the plane equation when it is expressed in the standard form. The normal vector is used to define orientations, such as finding lines perpendicular or parallel to the plane.

Equation of a Line in Space

A line in three-dimensional space can be represented parametrically by a point and a direction vector. The direction vector indicates the line's orientation and is crucial when defining lines that are perpendicular to planes since it can be aligned with the normal vector of the plane.

Perpendicularity between a Line and a Plane

A line is said to be perpendicular to a plane if its direction vector is parallel to the normal vector of the plane. This relationship is central when determining the equation of such a line, as it allows the direct use of the plane’s normal vector as the direction vector for the line.

Intersection of a Line and a Plane

The point of intersection between a line and a plane is found by substituting the parametric equations of the line into the equation of the plane. This method yields a parameter value which, when substituted back into the line's equations, gives the coordinates of the intersection point.

Distance from a Point to a Plane

The distance from a point to a plane is calculated using the formula that employs the plane's normal vector. It involves taking the absolute value of the plane equation evaluated at the point, divided by the magnitude of the normal vector. This formula provides a measure of the shortest distance from the point to the plane.

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