## 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.

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## 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 (x_{1}, y_{1}) and (x_{2}, y_{2}), 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.