There are many concepts in Mathematics that students find it simple to work with. However, when try to solve a question using the same concept, they end up with a wrong solution. I thought to pick one and explain more in detail : Distance (sounds like di·stns 😊) Formula.
The distance formula is one of the many useful mathematical equation, used to calculate the distance between two points in a coordinate plane or in the complex plane. It is based on the Pythagorean theorem and can be applied in two-dimensional (2D) , three-dimensional (3D) spaces and in the complex planes.
In a 2D coordinate plane, the distance formula calculates the distance between two points (X1, Y1) and (X2, Y2) as follows:
d = √[(X1 – X2)² + (Y2 – Y1)² ]
Here, d represents the distance between the two points.
Example : Lets assume point P1(2,3) and P2(4,5)
Here, we have:
x1 = 2, y1 = 3, x2 = 4, y2 = 5.
Therefore, we have:
(X2 – X1)² = (4-2)² = 4
(Y2 – Y1)² = (5-3)² = 4
distance (d) = √(4 + 4) = 2√2 units.
You can now see the FUN use of distance formula (wink! 😉)
In a 3D coordinate space, the distance formula computes the distance between two points (x1, y1, z1) and (x2, y2, z2) using the following formula:
d = √[( X2 – X1)² + ( Y2 – Y1)² + ( Z2 – Z1)²]
Similarly, d represents the distance, while (x1, y1, z1) and (x2, y2, z2) indicate the coordinates of the two points.
Example : Lets assume point P1(2,3,4) and P2(4,5,6)
Here, we have:
x1 = 2, y1 = 3, z1 = 4 x2= 4, y2= 5,z2=6.
Therefore, we have:
(x2 – x1)² = (4-2)² = 4
(y2 – y1)² = (5-3)² = 4
(z2 – z1)² = (6-4)² = 4
distance (d) = √(4 + 4 + 4) = 2√3 units.
In the Complex Plane , The modulus of the complex number a + bi is
∣a + bi∣ = √a2 + b2.
This is the distance between the origin (0, 0) and the point (a, b) in the complex plane. For two points in the complex plane, the distance between the points is the modulus of the difference of the two complex numbers.
Let (a, b) and (s, t) be points in the complex plane. The difference of the complex
numbers is
(s + ti) − (a + bi) = (s − a) + (t − b)i.
The modulus of the difference is
∣(s − a) + (t − b)i∣ = √(s − a)² + (t − b)².
Example : Let a + bi = 2 + 3i and s + ti = 5 − 2i.
The difference between the complex numbers is
(5 − 2i) − (2 + 3i) = (5 − 2) + (−2 − 3)i = 3 − 5i.
The distance is
d = √3² + (−5)² = √34 ≈ 5.83 units
By utilizing the distance formula, we can determine the straight-line distance between two points, treating it as the hypotenuse of a right triangle formed by the difference in their coordinates. This formula is heavily employed in various fields, including mathematics, physics, engineering, and computer science.
Cheers! ❤️
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