Distance Formula in 3 Dimensions
The distance formula in three dimensions is used to find the distance between two points in a 3D coordinate system
The distance formula in three dimensions is used to find the distance between two points in a 3D coordinate system. It is an extension of the distance formula in 2D.
The formula is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In a 3D space, let’s denote two points as P(x1, y1, z1) and Q(x2, y2, z2). The distance between these two points can be calculated using the distance formula:
d = √((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2)
This formula accounts for the differences in the x, y, and z coordinates of the two points. We square the differences, add them up, and then take the square root of the sum to get the distance.
To understand the formula better, let’s consider an example:
Example:
Consider the points P(1, 2, 3) and Q(4, 5, 6). We want to find the distance between these two points.
Using the distance formula:
d = √((4 – 1)^2 + (5 – 2)^2 + (6 – 3)^2)
= √(3^2 + 3^2 + 3^2)
= √(9 + 9 + 9)
= √27
≈ 5.196
So, the distance between points P(1, 2, 3) and Q(4, 5, 6) is approximately 5.196 units.
The distance formula in three dimensions is crucial in various fields such as physics, engineering, computer graphics, and many applications of 3D modeling. You can use it to find the distance between any two points in a 3D coordinate system.
More Answers:
Calculating the Right Riemann Sum for ∫√(831+x) dx: Understanding and Applying the Riemann Sum FormulaAn in-depth explanation of finding the derivative of an integral and the application of the chain rule
Understanding the Derivative of the Integral Function using the Fundamental Theorem of Calculus: Explained with Examples and Step-by-Step Derivation