Ray
GeometryA ray is a part of a line that has one endpoint and extends infinitely in one direction.
Definition
A ray starts at one endpoint and goes on forever in one direction. Think of a ray of sunlight - it starts at the sun and keeps going in one direction. We draw an arrow on the end that goes on forever.
Example
A flashlight beam is like a ray - it starts at the flashlight and goes on indefinitely in one direction. Ray $AB$ starts at point $A$, passes through point $B$, and continues past $B$ forever.
Key Insight
Rays are important for understanding angles. An angle is formed by two rays that share the same starting point (called the vertex). You literally need rays to describe what an angle is.
Definition
A ray is a subset of a line that starts at a fixed endpoint and extends infinitely in one direction. Ray $AB$ (written with a one-directional arrow above $AB$) starts at $A$, passes through $B$, and extends beyond $B$ without end. The starting point is the endpoint (or origin) of the ray.
Example
Ray $AB$ and ray $BA$ are different: ray $AB$ starts at $A$ and goes through $B$, while ray $BA$ starts at $B$ and goes through $A$ - they point in opposite directions. Two opposite rays form a straight line.
Key Insight
Two rays with a common endpoint form an angle. The endpoint is the vertex of the angle and the two rays are its sides. This connection makes rays essential vocabulary for angle measurement and classification.
Definition
A ray from point $A$ through point $B$ in $\mathbb{R}^n$ is the set $\{A + t(B - A) : t \ge 0\}$. It is a half-line: the translation of the non-negative real half-line into the ambient space. Two opposite rays from a common point partition the line through that point into two half-lines.
Example
In $\mathbb{R}^2$, the ray from origin in direction $(\cos\theta, \sin\theta)$ is $\{(t\cos\theta, t\sin\theta) : t \ge 0\}$. Polar coordinates use such rays from the origin. The concept generalizes to half-spaces in higher dimensions.
Key Insight
Rays are used to define angles in terms of directed rotation in the plane. In complex analysis, a ray from the origin in the complex plane corresponds to fixing the argument (angle) of a complex number, connecting geometric rays to analytic functions.