Question

To determine a plane in 3D space, you typically need at least three non-collinear points or some other information that allows you to define the plane.

In this case, you have a point P(10,−2,1) and a line with the parametric equations:
3x−7=6y+8=5z+4
This line can be represented as L=(7+3t,−8+6t,−4+5t).

Answer 1

A plane in 3D space requires three non-collinear points or equivalent information for definition. Given a point and a parametric line, additional information is needed to define the plane. This is answered using concepts from the Cartesian coordinate system and vector representations.

Step-by-step explanation:

To define a plane in 3D space, you need at least three non-collinear points or equivalent information. Given a point P(10,-2,1) and a line L with parametric equations 3x-7=6y+8=5z+4, we can rewrite the line in vector form as L=(7+3t,-8+6t,-4+5t).

In three-dimensional space, positions are given by three coordinates in the Cartesian coordinate system, namely x, y, and z. These are defined by the unit vectors i, j, and k. A line or a vector in space can be described using parametric or vector equations, providing a direct way to express movement or direction.

The Cartesian coordinate system forms the basis for vector representation in space. The concept of describing motion involves these coordinates as functions of time, thus allowing for the determination of positions and paths of particles or points in space. In this case, to completely determine the plane created by the point P and line L, additional information would be required, such as a second point on the plane or a normal vector to the plane.