Why can a line and a plane be named in more ways than one?
A line and a plane can be named in more ways than one because there are different ways to describe their properties, orientation, and position in space.
1. Point and direction vector(s): A line can be named by specifying a point on the line and providing a direction vector that indicates how the line extends in space. This is known as the point-direction form. For example, a line passing through the point (2, 3, 1) with a direction vector of <1, -2, 0> can be represented as (2, 3, 1) + t<1, -2, 0>, where t is a parameter that allows us to generate different points along the line.
2. Two points: A line can also be named by specifying two distinct points on the line. This is known as the two-point form. For example, a line passing through the points (1, 2, 3) and (4, 5, 6) can be represented as (1, 2, 3) + t(4, 5, 6) - (1, 2, 3), where t is a parameter that generates points between the two given points.
3. Normal vector and a point: A plane can be named by providing a normal vector perpendicular to the plane and a point that lies on the plane. This is known as the normal-point form. For example, a plane with a normal vector of <2, -1, 3> and passing through the point (1, 2, 3) can be represented as 2x - y + 3z = 2(1) - 1(2) + 3(3), using the equation of a plane. This form allows us to describe the plane uniquely.
4. Three non-collinear points: A plane can also be named by specifying three non-collinear points that lie on the plane. For example, a plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9) can be represented as the set of all points that are collinear with these three given points. This form defines a unique plane.
In summary, there are different ways to name a line or a plane based on their properties, orientation, and position in space. The choice of naming convention depends on the information available and the desired representation.