March 24, 2021

Chapters

In this article, we have jotted down various distance formulas. These formulas will help you to solve distance related problems easily. So, let us get started.

## Distance Between Two Points in a Plane

Distance between two points A and B tells us the shortest distance between these two points. Two points A and B along with their coordinates are shown in the figure below:

Consider the following example:

Calculate the distance between point A (5, -2) and point B (8, 3).

Use the following formula to calculate the distance between these two points:

## Distance from a Point to a Line in a Plane

Distance from a point to a line is the shortest distance between a line and point P. In the following figure, you can see a line r and point P. The shortest distance between point P and line r is depicted by a perpendicular line .

## Distance to the Origin in Plane

Consider the following example:

Calculate the distance between the line r whose equation is 2x + 5y + 2 = 0 and origin O.

Use the following formula to compute the distance between the line r and origin O.

## Distance between Parallel Lines in a Plane

The distance between two parallel lines in a plane shows the shortest distance between two lines in a plane. The following figure shows two parallel lines r and s in a plane.

Consider the following example:

Calculate the distance between two parallel line r which has an equation 3x + 5y + 1 = 0 and line s which has an equation 3x + 5y + 7 = 0.

Use the following formula to compute the distance:

Substitute the values in the formula like this:

## Distance Between Two Points in a Space

The minimum distance between two points in a three dimensional space is calculated using the following formula:

Consider the following example:

Calculate the distance between two point A (3, 4, 5) and B (7, 1, 4).

Use the following formula to calculate the distance:

Substitute the values in the above formula:

## Point-Line Distance - Space

The distance from a point, **P**, to a line, **r**, tells us the minimum distance from the point to one of many points on the line.

The formula to compute the distance between a line r and the point P is given below:

Here, reflects the direction vectors and reflects the distance between the point P and point A on the line.

Consider the following example:

=

Now, we will make 2 groups. One group will contain all the values that have t and the other group will consist of all the values that do not have t like this:

= (i + 3j + 4k) + (2ti + 1tj + 1tk)

Now, we will factor out t from the second group like this:

=

The coefficients in the first group show the point from which the line passes through and the coefficients in the second group reflect the direction vectors.

Hence, the direction numbers for the line are and the coordinates of the point through which the lines passes are .

We have to calculate the distance between the point P (1 , 2, 5) and point on the line A (1, 3, 4).

Since have both A and P, hence we can easily calculate like this:

Now, we will find the cross product of as shown below:

Put these values in the following formula to get the distance:

## Distance between a Point and a Plane in a Space

Consider the following example:

Calculate the distance between a point P (3, 6, 7) and the plane

Use the following formula to calculate the distance:

Substitute the values in the above formula: