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Ray in Geometry

Just like a line a ray is also infinite in length so it cannot be measured. However, we can measure the distance between any two points on a ray. Before we start discussing ray in detail, let's have a look what line stands for in geometry. 

Line: A collection of all points lying along a straight path in a plane is called a line. A line has no endpoints. It extends infinitely in both directions. A line is named by any two points lying on the line. For example, see the picture below:

Line Segment

The above picture is of line $AB$. In mathematical terms, it is written like this: $\overleftrightarrow{AB}$ with arrow heads on both the sides. Note that the line has no endpoints.

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Definition

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Ray: A ray is a portion of a line that is exactly one end point. It extends infinitely in the other direction. A ray is also named by any two points lying on it. However, ray $AB$ is not same as ray $BA$. See picture below:

Ray

Note that ray $AB$ and $BA$ are not the same. They are opposite in direction to each other. Symbolically, ray $AB$ would be written as: $\overrightarrow{AB}$ and the ray $BA$ would be written as: $\overrightarrow{BA}$ or $\overleftarrow{AB}$.

Notating and Measuring a Ray

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For notating a ray, it is imperative to first find out in which direction is the ray going. Say for example a ray looks as follows:

Notating a Ray
 
In the above picture, the ray is moving from $H$ towards $G$, so we’ll call it ray $\overrightarrow{HG}$ or $\overleftarrow{GH}$. The other ray would be $\overrightarrow{AB}$. 

If there are more than two points on a ray, then each combination could mean a different ray. Say, for example, see the following picture:
 
Measuring a Ray

In the above picture, we see that rays $BA$ and $BC$ are in opposite direction. Rays $AC$ and $BC$ are in the same direction but yet they do not represent the same ray. Both have a different starting point. So they are different rays. Whereas, in the case of lines, line $AB$, line $BC$ or line $AC$ all three would represent the same line. This is how notations and representations of rays are different from lines.

Like stated earlier, one cannot measure the length of a ray, the, however, angle between two rays can be measured using a protractor. Consider two rays having a common end point. They would look like this:

Angle between Two Rays
 
The above picture has two rays $AB$ and $AC$ in different directions. The angle made by them can be measured using a protractor as shown below:

Two rays AB and AC in different directions

Basic Rules for Rays

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Most of the rules for rays have already been stated above. However, let us summarize them here:


1. All rays have exactly one end point.

2. All rays extend indefinitely in the other direction.

3. All rays can be named by any two points on the ray of which one is the end point.

4. The length of a ray is infinite and cannot be measured.

5. If two rays have a common end point, then the angle between them can be measured.

6. A section of a ray with two end points becomes a line segment. 

7. If two rays intersect in a third point, then we’d get two line segments as shown in the figure below:

Basic Rules for Rays
 
In the above picture, the two rays $LN$ and $JO$ intersect in the point $P$. That gives us two line segments $LP$ and $JP$ whose lengths can be measured.

8.  Two intersecting lines, generate four rays. This can be seen in the following figure:
 
Generating Rays from Two intersecting lines

Lines $AB$ and $CD$ intersect in point $O$. So now we have four rays: $OA$, $OB$, $OC$, and $OD$. As is evident, $OC$ and $OD$ are opposite rays and so are $OA$ and $OB$.

9.  For writing an equation of a ray on a coordinate plane, all we need to do is first write the equation of the corresponding line and then restrict the domain at one end. This we shall see when we do examples.

10.  Rays are also used in three-dimensional co-ordinate geometry and vector analysis. This is evident from the following figure.

Ray in three-dimensional co-ordinate geometry and vector analysis
 
Rays are commonly used to represent vectors. In the above figure, the ray $V_0$ $V_1$ represents the vector $u$. The ray $V_0\ V_2$ represents the vector $v$. These two rays are coplanar. That means, they lie in the same plane $P$. The ray $V_0\ n$ is perpendicular to the plane $P$. So we say that it is a normal vector to the plane $P$. The ray $P_1\ P_0$ does not line in the plane $P$. It is in a completely different plane with respect to the other rays.

Examples

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Example 1: Write the equation of the following rays:

Examples on Ray in Geometry
 
1. $\overrightarrow{BA}$

2. $\overrightarrow{BD}$

3. $\overrightarrow{CD}$

Solution:

1. $\overrightarrow{BA}$

The coordinates of point $B$ are (-2,-2) and those of A are (2,3). So let, 

($x_1$,$y_1$) =  (-2,-2)

And 

($x_2$,$y_2$) =  (2,3)

Then the slope of the line BA can be given  by the formula:

$m$ = $\frac{y_2\ -\ y_1}{x_2\ -\ x_1}$

Substituting the values into the above formula we have:

$slope$ = $m$ = $\frac{3\ -\ (-2)}{2\ -\ (-2)}$

Simplifying and evaluating that we have:

$m$ = $\frac{3\ +\ 2}{2\ +\ 2}$

$m$ = $\frac{5}{4}$

Using the point-slope formula we have the equation of the line $BA$ as:

$y\ -\ y_1$ = $m(x\ -\ x_1)$

$y\ -\ (-2)$ = $\frac{5}{4}$$(x\ -\ (-\ 2))$

$y\ +\ 2$ = $\frac{5}{4}$ $(x\ +\ 2)$

$y\ +\ 2$ = $\frac{5}{4}$ $x\ +$ $\frac{5}{2}$

$y$ = $(\frac{5}{4})$ $x\ +$ $\frac{5}{2}$$-\ 2$

$y$ = $\frac{5}{4}$ $x\ +$ $\frac{1}{2}$

This is the equation of the line $BA$. Now if we restrict the domain such that the x values cannot be lesser than $-2$, then we get the required equation of the ray $BA$. Thus our final answer would be:

$y$ = $\frac{5}{4}$ $x\ +$ $\frac{1}{2}$,$x\ \geq\ -\ 2$

2. The equation of the ray $BD$ would just be $y$ = $-2$. The domain restriction here would be same as previous one which is $x\ \geq\ -2$.

3. The equation of the ray $CD$ would be again same as before, $y$ = $-2$. However this time the domain restriction would be: $x\ \geq\ 1$. This is evident from the graph.

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