Vertical Line – Definition, Equation, Facts, Examples, FAQs

Home » Math Vocabluary » Vertical Line – Definition, Equation, Facts, Examples, FAQs

What Is a Vertical Line in Math?

Vertical line is a line perpendicular to a horizontal line or a surface. In a coordinate plane, the y-axis is a vertical line since it is perpendicular to the x-axis, the baseline. Thus, any line that is parallel to the y-axis is a vertical line. A vertical line is also called a standing line.

Examples of vertical lines:

Vertical lines

Vertical Line Definition

A vertical line is a straight line that runs from the top to the bottom, parallel to the y-axis, on the Cartesian coordinate system. It has an undefined slope and an equation of the form x = c, where “c” represents a constant value, indicating that the x-coordinate remains constant for all points on the line. Vertical lines divide the coordinate plane into two halves and have no y-intercept.

Properties of Vertical Lines

  1. Vertical lines are parallel to the y-axis. Thus, they do not intersect the y-axis. In other words, they do not have a y-intercept.
  1. The equation of a vertical line is of the form of x = a, where a is the x intercept.
  1. Since there is no change in the x-coordinates in a vertical line (x is constant), the denominator of the slope is always zero. Thus, the slope of a vertical line is undefined.
  1. A vertical line intersects a horizontal line at a right angle.

Equation of a Vertical Line

The equation of a vertical line in math can be given by x = ± a, where x is the x-coordinate and (a,0) is the point where the line intersects the x axis, the x-intercept.

For example, consider the line which intersects the x-axis at (5,0) and is parallel to the y-axis. We get the equation of the line as x = 5.

Vertical Line on a Coordinate Plane

On a coordinate plane, the x-axis is considered as the horizontal axis, whereas the y-axis is the vertical axis. Vertical line in a coordinate plane is a line that is parallel to the y-axis. 

For a vertical line, all the points lying on the line have the same x-coordinate. Thus, the x-coordinate stays the same, only the y-coordinate changes. A vertical line never intersects the y-axis. 

Vertical line x = 5 in Cartesian plane

Slope of a Vertical Line

Slope of a line tells us how steep a line is. If a line passes through two points having coordinates (x1, y1) and (x2, y2), then we can define the slope of a line as

Slope $= m = \frac{y_{2} \;-\; y_{1}}{x_{2}\;-\; x_{1}} = \frac{\Delta y}{\Delta x} = \frac{Change \;in\; y}{Change \;in\; x} = \frac{Rise}{Run}$

As mentioned earlier, the x-coordinate remains constant in the equation of a vertical line. Only the y-coordinate changes. Thus, the denominator in the formula of the slope becomes 0. Since the division by 0 is not defined, the slope of a vertical line is undefined.

Slope of a vertical line is undefined

Difference between Horizontal Line and Vertical Line

  Horizontal LineVertical Line
It is parallel to the x-axis.It is parallel to the y-axis.
It is perpendicular to the y-axis.It is perpendicular to the x-axis.
The equation is of the form y = ± a.The equation is of the form x = ± a.
The slope of a horizontal line is 0.The slope of a vertical line is undefined.
Vertical Line – Definition, Equation, Facts, Examples, FAQsVertical Line – Definition, Equation, Facts, Examples, FAQs

Vertical Line Test

Vertical line test is used to assess whether a given graph represents a function or not. It is based on the fact that a function can only have one output for every input. This test ensures that for each input (x-value), there is only one corresponding output (y-value), satisfying the criteria of a function.

If in a graph, a vertical line has more than one intersection, it means that there is more than one output for a single input, which means it cannot be a function. 

Thus, any vertical line in the plane can only ever cross the graph of a function once.

Vertical line test examples:

Vertical Line Test

Vertical Lines of Symmetry

When a vertical line goes through the middle of a shape from top to bottom dividing it into two identical halves, that vertical line is called a vertical line of symmetry.

Vertical lines of symmetry

Facts about Vertical Lines

  • The slope of a vertical is always undefined.
  • A vertical line always goes from top to bottom making a 90 degree angle with the x axis.
  • A vertical line makes a right angle with the x-axis.

Conclusion

In this article, we learned about vertical lines, their meaning and definition in geometry, their properties, equation and slope. Let’s look at a few examples. 

Solved Examples on Vertical Lines

1. Find the slope of the line x = – 3.

Solution:

Since x = – 3 is a line parallel to the y-axis. 

Thus, it is a vertical line.

The slope of this line is undefined.

2. Find the equation of the vertical line passing through the point (2, 4).

Solution:

The equation of the vertical line is of the form x = ± a, where a is the x-intercept.

Since the line passes through (2, 4), the x-intercept is 2.

Thus, the equation of the vertical line is x = 2.

We can also write it as x – 2 = 0.

3. Find the equation of the line given in the figure

Vertical line x =-3 on graph

Solution:

The line is parallel to the y axis and passes through the point ( 3, 0) on the x axis.

Thus, the x-intercept is 3.

The equation of the given vertical line is x = 3.

4. Plot the line x = 1.5 on a graph.

Solution:

x = 1.5 represents a vertical line passing through the point (1.5, 0). 

Vertical line x = 1.5

Practice Problems on Vertical Lines

Vertical Line - Definition, Equation, Facts, Examples, FAQs

Attend this quiz & Test your knowledge.

1

The slope of the line $x = 10$ is

m = 9
m = 2
m = 0
m = undefined
CorrectIncorrect
Correct answer is: m = undefined
Since x = 10 gives us a line parallel to the y axis, the slope of this line will be undetermined.
2

A vertical line is parallel to the _____.

x-axis
y-axis
line x = y
None of the above
CorrectIncorrect
Correct answer is: y-axis
A vertical line is parallel to the y-axis.
3

Which of the following equations represent a vertical line?

x = 2.5
y = 2.5
y = x + 2.5
x = 2.5 + 2.5y
CorrectIncorrect
Correct answer is: x = 2.5
The equation x = 2.5 represents the vertical line.

Frequently Asked Questions about Vertical Lines

Lines which are parallel to the x-axis and make a right angle with the y-axis are called horizontal lines.

The slope of a horizontal line is 0.

The up and down direction is the vertical line.

A vertical line is drawn in the up and down direction, from top to bottom on a page or a shape. It intersects a horizontal line at a right angle.