Distance Between Two Points

Distance is an important parameter used in day-to-day life. We can easily tell how far two objects are placed by measuring their distance. Distance is the length of the straight line that joins the two points given in space. 

With the help of the ‘distance formula’, we can easily calculate the distance between two well-defined points in a particular coordinate system. The distance formula is a general expression with the help of which we can calculate the length of a straight line joining two points: P(a1,b1) and Q(a2,b2) (that is, the distance between P and Q).

How to Define the Distance between Two Points?

We can define the distance between two points, say P and Q, as the length of a straight line joining points P and Q. Distance is also defined as the magnitude of the separation vector of points P and Q defined concerning a particular coordinate system. Thus, if we know the position vectors or coordinates of two points, we can calculate the distance between them.

Formula to Calculate the Distance between Two Points  

1. Two-dimensional cartesian coordinate systems

Let us consider a particular cartesian coordinate system with O(0,0) as the origin. Let P(a1,b1) and Q(a2,b2) be the two points separated by the distance D, then D is given as 

D=(a1-a2)2+(b1-b2)2

2. Plane polar coordinate system

If we consider two points P(r1,𝚹1) and Q(r2,𝚹2), then the distance (D) between P and Q is given as:  

D=r12+r22-2r1r2cos(𝚹1-𝚹2)

3. Three-dimensional cartesian coordinate system

           In three-dimensional cartesian coordinate system, if two points E(a1,b1,c1) and F(a2,b2,c      ) are given, then the distance D between them is given as: 

D=(a1-a2)2+(b1-b2)2+(c1-c2)2

Derivation of Distance Formula 

4. Cartesian coordinate system

Consider a two-dimensional cartesian coordinate system. Let O(0,0) be the origin. Consider two points P(a1,b1) and Q(a2,b2). 

As shown in the diagram, Point R will have the coordinates (a1,b2). We want to calculate the length of segment PQ. 

Consider the right-angled triangle PRQ. According to the Pythagoras theorem: 

|PQ|2=|PR|2+|RQ|2 …………….(1)

From diagram: 

|PR|=|b1-b2| and |RQ|=|a1-a2|

Substituting the values of |PR| and |RQ| in the equation (1), we get:

D2=(b1-b2)2+(a1-a2)2

∴ D=(b1-b2)2+(a1-a2)2

This is the required distance formula. 

Qus. Calculate the distance between the points E(2,4) and F(-1,7).

Ans. Points E and F are given in cartesian coordinates. Using equation (2), we can write:

 D=(b1-b2)2+(a1-a2)2

 D=(4-7)2+(2-(-1))2

 D=(-3)2+(3)2=9+9=18 units

This is the distance between points E and F

  5. Plane polar coordinates 

Consider the plane polar coordinate system as shown in the diagram below. Let O(0,0) be the origin. Let us consider the two points, L(r1,𝚹1) and M(r2,𝚹2), as shown in the diagram.

The radial distance of point L from the origin is R1, and the angle made with the x-axis is 𝚹1

The radial distance of point M from the origin is R2, and the angle made with the x-axis is 𝚹2

The angle between the position vectors of L and M is (𝚹1-𝚹2).

We want to find the length of segment LM. Let the length of LM is equal to R.

According to the law of cosine, we can write:

R2=R12+R22-2R1R2cos(𝚹1-𝚹2)

∴ R=R12+R22-2R1R2cos(𝚹1-𝚹2) …………….(2)

We can calculate the distance between any two points in the plane’s polar coordinates with the help of equation (2).

Qus. Calculate the distance between the points E(5,𝛑/3) and F(10,𝛑/6).

Ans. Points E and F are given in plane polar coordinates. Using equation (2), we can write.

 R=52+102-2(5)(10)cos(𝛑/6-𝛑/3)

 R=25+100-100cos(-𝛑/6)

 R=125-86.60=38.4=6.20 units

This is the distance between points E and F.

Properties of the Distance between Two Points

1. Distance is a scalar quantity, whereas displacement is a vector quantity. Distance is the magnitude of the displacement vector joining two points. 

2. Distance between two points is always greater than or equal to zero.

3. If the distance between two points is zero, the two points are coinciding (then it is the same point, that is, (a1,b1)=(a2,b2)).

4. The distance cannot be negative. 

5. The distance cannot be a complex number.

6. Physical units of measuring distances are generally metres, feet, inches, yards, parsec, nautical miles, miles, etc.