Vertical line is a line that is perpendicular to the base of any geometrical object and generally, we state the bottom of the object as a base. In simple words, we define a vertical line as a line that is perpendicular to the horizontal line. In context to the cartesian coordinate system, vertical lines are defined as lines that are parallel to the y-axis or perpendicular to the x-axis. A vertical line always goes from top to bottom and is also called a standing line.
Various examples where we observe vertical lines are the lines joining the base of the rectangle, square, etc. The vertical lines are very useful for solving and explaining various geometrical problems. This article explores the topic of Vertical Lines including its subtopics like its definition, diagram, relation with other lines, etc. So, let’s learn about the vertical lines in detail in this article.
Vertical Line Definition
We define a vertical line as a line in which all points have the same x-coordinate. In other words, a line that is perpendicular to the x-axis and parallel to the y-axis is called Vertical Line. In real life, we observe various examples of vertical lines such as a long tower, the legs of a table and a chair, a long tree, etc.
The slope of the vertical line is undefined as it makes a 90° angle with the x-axis. The vertical line goes from top to bottom in the Cartesian plane. The image added below shows the vertical line,
Vertical Line on Coordinate Plane
Coordinate Plane is a plane that is formed by the intersection of the x-axis and the y-axis. So any line parallel to the y-axis is called the vertical line. A vertical line has a fixed x-coordinate and a variable y-coordinate. The general point on a vertical line is (c, b) where c is constant and the value of b changes accordingly.
At the x-axis, the vertical line has the coordinate (c, 0) and it is perpendicular to the x-axis.
Equation of Vertical Line
The equation of the vertical line is given as,
x = h
Where h is any real constant.
All the points in these lines have the coordinates as (h, b) where the h is constant and b is variable.
For the equation of the vertical line i.e., x = 11. We can say that this line passes through the point (11, 0) on the x-axis and various points on this vertical line are, (11, -3) and (11, 9), (11, 8/9), etc.
Example of Vertical Lines
There are various scenarios where we see vertical lines in real life and some of them are we see vertical lines at the corner of the building these lines run across the height of the building, and the height of a tree or a mountain is measured using a vertical line. In coordinate geometry, various examples of the vertical line are, x = 9 is a vertical line passing through points (9, 0), (9, -1), (9, -2), (9, 4), (9, 8), etc. This line cuts the x-axis at (9, 0) and is parallel to the y-axis.
Some other examples of vertical lines are,
- Equation of a vertical line through (-1, 0) is x = -1
- Equation of a vertical line through (8, 0) is x = 8
Slope of Vertical Line
For a vertical line, the slope of the line is undefined. This can be understood by the definition of the slope of the line as,
Slope of a line = Change in y-coordinate/ Change in x-coordinate
OR
m = (y2 – y1) / (x2 – x1)
For a vertical line, we know that the x-coordinate never changes, and thus x2 = x1 = x, so x2 – x1 = 0
⇒ m = (y2 – y1)/0 = Undefined
Thus, the slope of the vertical line is undefined.
Properties of Vertical Lines
Vertical lines have some special properties that include,
- Vertical line is a line that is always parallel to the y-axis.
- Vertical line is a line that is always perpendicular to the x-axis.
- Slope of the vertical line is undefined or infinity.
- Equation of the vertical line is of the form, x = h(constant).
Vertical and Horizontal Lines
There are key differences between both vertical lines and horizontal lines, some of these differences are listed in the following table:
| Parallel to the y-axis |
Parallel to the x-axis |
| Slope is undefined or infinite |
Slope is zero |
Equation of vertical line is,
x = constant value |
Equation of horizontal line is,
y = constant value |
| Extends from top to bottom |
Extends from left to right |
| Intersects the x-axis at one point |
Intersects the y-axis at one point |
| Graph of a vertical line is a straight line parallel to the Y-axis |
Graph of a horizontal line is a straight line parallel to the X-axis |
Equation of vertical lines is,
x = -2
x = 7, etc |
Equation of horizontal lines is,
y = 3
y = -6, etc |
| 90 degrees |
0 degree |
| 0 degree |
90 degrees |
Vertical Line Test
Vertical line test is a test that tells us whether a given graph is a function or not. Any relation is considered a function if any vertical line drawn along the graph of the relation intersects the graph only at one point. We know that for any relation to be considered a function it only has one output for every input. And thus, if the vertical line cuts the graph of the given relation more than once then it is not a function.
In the image added below, a vertical line drawn to y = f(x)(figure 1) cuts the graph at only one point, and thus, y = f(x) is a function because it follows the vertical line test. For instance, the graph of y = f(x)(figure 2) is not a function because a vertical line drawn cuts the graph at two points and it fails the vertical line test.
Vertical Line of Symmetry
A line running from the top to the bottom of any figure that divides the figure into two identical halves that are mirror images of each other is called the vertical line of symmetry. There are various figures in which the vertical line of symmetry is observed that include square, rectangle, circle, etc. The image added below shows the vertical line of symmetry of these figures,
A part of these the alphabet in the English language also shows a vertical line of symmetry. There are a total of 11 alphabets in the English language that shows the Vertical line of symmetry that includes, A H I M O T U V W X Y. The image added below shows the vertical line of symmetry for the same.
Vertical Line Summary
All the basic of the vertical lines can be summarized as,
- Vertical line is a straight line perpendicular to the baseline or the horizontal line.
- Equation of a vertical line is x = h (constant).
- A vertical line is always parallel to the y-axis.
- Slope of vertical lines is not defined.
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Vertical Line Examples
Example 1: Find the equation of the vertical line passing through the point (1, -1).
Solution:
Given point (1, -1)
Equation of the vertical line passing through a point (h, k)
x = h
Substituting the values in the above equation we get,
x = 1
Thus, equation of the vertical line passing through the point (1, -1) is x = 1.
Example 2: Find the equation of the vertical line passing through the point (5, 9).
Solution:
Given point (5, 9)
Equation of the vertical line passing through a point (h, k)
x = h
Substituting the values in the above equation we get,
x = 5
Thus, equation of the vertical line passing through the point (5, 9) is x = 9.
Example 3: Find the equation of the vertical line when the x-intercept of the line is 5.
Solution:
Equation of the vertical line is,
x = h
where h is x-intercept
Given
Equation of the vertical line,
x = 5
Thus, the equation vertical line with x-intercept as 5 is, x = 5
Example 4: Find the equation of the vertical line when the x-intercept of the line is -11/3.
Solution:
Equation of the vertical line is,
y = k
where h is x-intercept
Given
Equation of the vertical line,
x = -11/3
3x = -11
3x + 11 = 0
Thus, the equation vertical line with x-intercept as -11/3 is, 3x + 11 = 0
FAQs on Vertical Lines
1. What are Vertical Lines?
Vertical lines are lines that are parallel to the y-axis. They run top to bottom in the Cartesian system.
2. What is the Equation of the Vertical Line?
Equation of the vertical line is,
x = h
where h is the intercept on the x-axis.
3. What is the slope of a Vertical line?
The slope of the vertical line is undefined or infinity. The vertical line makes an angle of 90 degrees with the x-axis or the horizontal line.
4. What are Examples of the Vertical Lines?
Examples representing the verticle lines are,
- Cornes of any Buildings
- Height of any Figure, etc.
5. What are the Vertical Lines on the Globe called?
Vertical lines running on the globe are called Longitude and they run parallel to the Prime Meridian.
6. What are the Properties of Vertical Lines?
Various properties of the Vertical lines are,
- They are parallel to the y-axis.
- They are perpendicular to the x-axis.
- The slope of the vertical line is infinity, etc.
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