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:
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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
- 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.
- The equation of a vertical line is of the form of x = a, where a is the x intercept.
- 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.
- 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.
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.
Difference between Horizontal Line and Vertical Line
| Horizontal Line | Vertical 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 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 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.
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
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).
Practice Problems on Vertical Lines
Vertical Line - Definition, Equation, Facts, Examples, FAQs
The slope of the line $x = 10$ is
Since x = 10 gives us a line parallel to the y axis, the slope of this line will be undetermined.
A vertical line is parallel to the _____.
A vertical line is parallel to the y-axis.
Which of the following equations represent a vertical line?
The equation x = 2.5 represents the vertical line.
Frequently Asked Questions about Vertical Lines
What are horizontal lines?
Lines which are parallel to the x-axis and make a right angle with the y-axis are called horizontal lines.
What is the slope of a horizontal line?
The slope of a horizontal line is 0.
Which direction is considered as vertical?
The up and down direction is the vertical line.
How can we identify a 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.