Distance Between Two Points in Complex Plane

Consider some complex numbers such as (3 + 5i), (– 2 + 5i), (0 + 10i), (4 + 2i), (– 3 –3i) and (2 – 1i), which correspond to the ordered pairs (3, 5), ( – 2, 5), (0, 10), (4, 2), (–3, –3), and (2, – 2) respectively. They are represented geometrically by the points A, B, C, D, E, and F, respectively. So, in the argand plane, the modulus of the complex number is

(x + i.y) = √(x2 + y2) 

The above expression is the distance between the point P(a, b) and the origin O (0, 0). In the following article, we will find the formula for the distance between two points in a complex plane.

Formula for the distance between two points in complex plane

According to distance between two points in complex plane notes, modulus of the complex numbers (a + bi) is given as, 

∣x + y.i∣ =√( x2 + y2 )

The above expression gives the distance between the origin (0, 0) and the point (x, y) in a complex plane. For two given points in a complex plane, the distance between the points is defined as the modulus of the difference between the two complex numbers. 

Let’s consider (x, y) and (p, q) are two points in a complex plane, 

The difference of the complex numbers is given by, 

(p + qi) – (x + yi),

(p – x) + (q – y)i. 

The modulus of the difference is given by,

 ∣ (p − x) + (q − y)i∣ 

 p – x2 +q – y2

So, 

d = √((p – x)2+(q – y)2  )

where d is the difference between the two points in a complex plane.

The midpoint of a line segment in the complex plane

The following formula gives the midpoint of a line segment in a complex plane,

Midpoint = {x + p2} + {y + q2.i}

Solved Examples

Example 1: Find the distance between the points (3 + 2i) and (2 − 5i) in a complex plane. 

Solution: Let (x + yi) = (3 + 2i)

and (p + qi) = (2 − 5i) 

The difference between the complex numbers is given by, 

(2 − 5i) − (3 + 2i), 

(2 − 3) + (−5 − 2)i, 

-1 − 7i. 

The distance is given by, 

d =  (-1)2 +(-7)2  

50 units.

Therefore, the distance between the given two points in a complex plane is 50   units.

Example 2: Find the midpoint of the line segment joining the two points (2−3i) and (2+4i).

Solution: Let, (x + yi) = (2 − 3i) 

and (p + qi) = (2 + 4i), 

Applying the Midpoint Formula, 

Midpoint = {x + p2} + {y + q2.i}

{(2 + 2)2} + {(-3) + 42.i}

2 + 0.5i 

The midpoint of the line segment joining the points (2 − 3i) and (2 + 4i) is (3 + 0.5i).

Example 3:  Find the distance between the points (6 + 4i) and (4 − 10i) in a complex plane. 

Solution: Let (x + yi) = (6 + 4i)

and (p + qi) = (4 − 10i) 

The difference between the complex numbers is given by, 

(4 − 10i) − (6 + 4i), 

(4 − 6) + (−10 − 4)i, 

-2 − 14i. 

The distance is given by, 

d = (-2)2 +(-14)2 ,

= 10 units.

Therefore, the distance between the given two points in a complex plane is 10 units.

Example 4: Find the midpoint of the line segment joining the two points (3−3i) and (2+2i)

Solution: Let, (x + yi) = (3 − 3i) 

and (p + qi) = (2 + 2i), 

Applying the Midpoint Formula, 

Midpoint = {x + p2} + {y + q2.i}

= {(2 + 3)2} + {(-3 + 2)2.i}

={52 + 12.i}

The midpoint of the line segment joining the points (3 − 3i) and (2 + 2i) is (52 + 12.i).

Example 5:  Find the distance between the points (7 + 4i) and (4 − 3i) in a complex plane. 

Solution: Let (x + yi) = (7 + 4i)

and (p + qi) = (4 − 3i) 

The difference between the complex numbers is given by, 

(4 − 3i) − (7 + 4i), 

(4 − 7) + (−3 − 4)i, 

-3 − 7i. 

The distance is given by, 

d = (-3)2 +(-7)2

58 units.

Therefore, the distance between the given two points in a complex plane is 58 units.

Example 6: Find the distance between the points (5 + 6i) and (2 + 5i) in a complex plane. 

Solution: Let (x + yi) = (5 + 6i)

and (p + qi) = (2 + 5i) 

The difference between the complex numbers is given by, 

(2 + 5i) − (5 + 6i), 

(2 − 5) + (5 − 6)i, 

-3 − 1i. 

The distance is given by, 

d = (-3)2 +(-1)2 ,

10 units.

Therefore, the distance between the given two points in a complex plane is 10 units.

Example 7: Find the midpoint of the line segment joining the two points (1−3i) and (1+2i)

Solution: Let, (x + yi) = (1 − 3i) 

and (p + qi) = (1 + 2i), 

Applying the Midpoint Formula, 

Midpoint = {x + p2} + {y + q2.i}

= {(1 + 1)2} + {(-3 + 2)2.i}

= (1 – 12i) 

The midpoint of the line segment joining the points (1 − 3i) and (1 + 2i) is (1 – 12i).

Conclusion

The above article gives the formula for the distance between two points in a complex plane and distance between two points in complex plane meaning. We also look into the formula for the midpoint of a line segment in a complex plane.

The distance between two points in a complex plane when two points are in an argand plane is given by,

d = p – x2 +q – y2 .

Here, (x + iy) and (p +iq) are two points in an argand plane.