Ray — Definition, Meaning & Examples
Worked Example
Problem: Point A is at coordinates (2, 3) and point B is at (5, 7). Describe ray AB and determine whether the point C = (8, 11) lies on it.
Step 1: Identify the endpoint and the direction. Ray AB starts at point A (the endpoint) and passes through point B, extending infinitely beyond B.
Endpoint: A=(2,3)
Step 2: Find the direction vector from A to B by subtracting coordinates.
AB=(5−2,7−3)=(3,4)
Step 3: Any point on ray AB can be written as A plus a non-negative scalar multiple of the direction vector.
P=(2+3t,3+4t)where t≥0
Step 4: Check if C = (8, 11) fits this form. Set 2 + 3t = 8, giving t = 2. Then check the y-coordinate: 3 + 4(2) = 11. Both coordinates match with t = 2 ≥ 0.
t=2≥0⇒C lies on ray AB
Answer: Ray AB starts at (2, 3), passes through (5, 7), and continues infinitely in that direction. Point C = (8, 11) does lie on ray AB because it is in the same direction from A as B, with t = 2.
Frequently Asked Questions
How do you name a ray?
A ray is named using two points: the endpoint is always listed first, followed by any other point on the ray. For example, ray AB (written AB) starts at A and passes through B. The order matters — ray AB and ray BA are different rays because they have different endpoints and may point in opposite directions.
What is the difference between a ray and a line segment?
A line segment has two endpoints and a finite length. A ray has exactly one endpoint and extends infinitely in one direction, so it has no measurable total length. A full line has no endpoints at all and extends infinitely in both directions.
Ray vs. Line Segment
A ray has one endpoint and infinite length in one direction. A line segment has two endpoints and a definite, finite length. Both are parts of a line, but a segment is bounded on both sides while a ray is bounded on only one side.
Why It Matters
Rays are fundamental in geometry for defining angles — every angle is formed by two rays that share a common endpoint (called the vertex). They also appear throughout physics and computer graphics, where "ray tracing" uses rays to model light paths. Understanding rays helps you work with coordinate geometry, trigonometry, and any situation involving direction from a fixed point.
Common Mistakes
Mistake: Writing the ray's name in the wrong order, such as BA when you mean the ray starting at A.
Correction: The endpoint must always come first in ray notation. AB starts at A; BA starts at B. These are generally two different rays.
Mistake: Thinking a ray has a finite length or two endpoints.
Correction: A ray has exactly one endpoint and extends without end in one direction. If it has two endpoints, it is a line segment, not a ray.