Distance is a mathematical quantity that is used to find out how far 2 points lie from each other in a 2D space. This parameter is perhaps one of the most important mathematical quantities. It is used to in advanced mathematics and physics to model such quantities as the velocity of a moving body, the magnitude and direction of electrical and gravitational forces, and in signal processing. Let us study the distance between two points in detail.

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## Distance Between Two Points

2D distance is the distance between 2 points in a 2D space. In a 2D space, each point, or location in the space, is qualified by 2 parameters like an x-coordinate, and a y-coordinate. We denote the combination of x-coordinate and y-coordinate in something known as an ordered pair, denoted by (x, y). Therefore, the coordinates of some point P would be represented as P(x,y).

**Browse more Topics under Introduction To Three Dimensional Geometry**

The 2D distance between two points is measured by the *distance formula*. Consider points P1 and P2, with coordinates given as P1(x_{1}, y_{1}) and P2(x_{2},y_{2}). Therefore, the distance between them would be denoted by d, where:

**d = √[( y _{2 }– y_{1})² + ( x_{1 }– x_{2})²]**

The way to think about distance calculation is in 3 parts:

- Calculate the difference in y-coordinates, and square the difference, denote as quantity q
_{1} - Calculate the difference in x-coordinates, and square the difference, denote as quantity q
_{2} - Take the square root of ( q
_{1 }+ q_{2 }).

### Example 1

Calculate the distance between the points P_{1}(1,2) and P_{2}(4,3)

Solution: In the given problem, we have: x_{1 }= 1, y_{1 }= 2, _{ }x_{2} = 4, and y_{2 } = 3. Therefore, we have:

q_{1 } = (3-2)² = 1

_{ }q_{2}= (4-1)² = 9

distance (d) = √(q_{1 }+ q_{2})

∴d = √(1 + 9) = √10 = 3.16227 units.**
**As you can see in the previous example, we have written the units of distance as ‘units’, and not ‘square units.’

### Example 2

Calculate the distance between the points P_{1}(1,2) and P_{2}(4,2)

Solution: In the given problem, we have: x_{1} = 1, y_{1} = 2, x_{2} = 4, and y_{2} = 2. Therefore, we have:

q_{1} = (2-2)² = 0

q_{2} = (4-1)² = 9

distance (d) = √(q_{1}+q_{2})

∴d = √(0 + 9) = √9 = 3 units.

#### Important Points:

- As you can see, in the above problem, we have y
_{2}-y_{1}= 0. This means that the points are on the same horizontal line. - The distance between any two points on the same horizontal line (where y
_{1}= y_{2}) is given by d = |x_{2}– x_{1}| where | | is used to denote the absolute value of x_{2}– x_{1}.

### Example 3

Calculate the distance between the points P_{1}(1,2) and P_{2}(1,5)

Solution: In the given problem, we have: x_{1 }= 1, y_{1 }= 1, x_{2} = 1, and y_{2 } = 2. Therefore, we have:

q_{1 } = (4-1)²= 9

q_{2 }= (1-1)² = 0

distance (d) = √(q_{1 }+ q_{2})

∴d = √(9 + 0) = √9 = 3 units.

#### Important Points:

- As you can see, in the above problem, we have x
_{1 – }x_{2}= 0. This means that the points are on the same vertical line. - The distance between any two points on the same vertical line (where x
_{1 = }x_{2}) is given by d = | y_{2 }– y_{1 }| where | | is used to denote the absolute value of y_{2 }– y_{1}

## Solved Question for You

**Question 1: Can the distance between any two points be negative?**

**Answer:** No, the distance between 2 points cannot be negative. This can be thought of in terms of 3 reasons:

- Distance is used to denote a physical quantity, expressing how far two points are from each other, and such a quantity cannot be negative.
- Distance is equal to the square root of the sum of two positive numbers. The sum of two positive numbers is always positive, and the square root of a positive number is always positive.
- In the exceptional case that the distance between 2 points is zero, it is still a non-negative value, hence the distance cannot be negative.

This concludes our discussion on the topic – distance between two points.

**Question 2: How to find Y-intercept?**

**Answer:** We can find the Y-intercept using the equation of the line. Put zero for the x variable and solve for y. Moreover, if the equation is written in the slope-intercept form then put in the slope and the x and y coordinates for a point on the line to solve for y.

**Question 3: What is the distance between two points?**

**Answer:** Basically, the distance between two points is the length of the line segment that connects them. Most importantly, the distance between two points is always positive and segments that have equal length are called congruent segments.

**Question 4: What is the formula for distance between two points?**

**Answer:** The formula of distance between two points is P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is given by: d (P, Q) = √ (x_{2} – x_{1}) + (y_{2} – y_{1}) 2. While the distance of a point is P(x, y) from the origin is given by d(0, P) = √ x_{2} + y_{2}.

**Question 5: What is the minimum distance?**

**Answer:** Minimum distance refers to the estimation of the statistical methods for fitting a model to data. Moreover, the closest pair of points problem, the algorithmic problems of two points that have the minimum distance among a larger set of points.

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