The locus of a point moving along a half-line

Understanding the Half-Line

A half-line, or ray, is a line that starts from a fixed point and extends infinitely in one direction.
The locus of a point moving along a half-line refers to the path this point follows as it moves from the fixed point, or the origin of the half-line, following the direction of the line to infinity.

The fixed start point of a half-line is often called the origin of the half-line.
The direction of the half-line is the way the line extends from the origin.
While a line exists along both directions from a point, a half-line exists in only one direction from the origin.

The equation for a half-line, or ray equation, from an origin (h, k) in the direction of a point (x, y) is given by (x - h) = t(a - h) and (y - k) = t(b - k), where t is a parameter greater than or equal to 0.
The loci of points on the half-line will satisfy these equations.

Geometrically, a half-line is an infinite line that starts from a fixed point. It is a one-dimensional figure.
A line segment has a fixed length and two endpoints, whereas a half-line has one endpoint and extends infinitely in one direction.
Using vector notation, a half-line can be written as OP + t(DP), where OP is the vector origin to point P and DP is the vector from D to P.

To understand the concept of a locus on a half-line better, draw diagrams and illustrate movement of points along the half-line.
Understanding how equations relate to the concept of loci is essential in grasping this topic. Review and practice with the ray equation.
Understanding the distinct properties of a line, line segment, and half-line can enhance understanding of loci on a half-line.
Making use of vector notation can simplify understanding of half-lines and their respective loci. Practice plotting half-lines using original points and directions given by vectors.