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## Distance Formula Definition

The distance formula is a crucial tool in analytic geometry, enabling you to calculate the distance between two points in a two-dimensional or three-dimensional space, given their Cartesian coordinates. It is fundamental for various mathematical computations and real-world applications.

### Understanding the Distance Formula

**The distance formula** derives its principles from the Pythagorean theorem. When you think of the distance between two points, you can visualise it as the hypotenuse of a right triangle formed by these points, thus connecting them geometrically through their coordinates.The formula for the distance between two points \(x_1, y_1\) and \(x_2, y_2\) in a two-dimensional plane is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

**Distance Formula:** The formula is given by \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\], where \(d\) is the distance between points \(x_1, y_1\) and \(x_2, y_2\).

Consider two points \(A = (3, 4)\) and \(B = (7, 1)\). The distance \(d\) between these points is calculated as follows:

- Subtract the x-coordinates: \(7 - 3 = 4\)
- Subtract the y-coordinates: \(1 - 4 = -3\)
- Square the differences: \(4^2 = 16\) and \((-3)^2 = 9\)
- Sum the squares: \(16 + 9 = 25\)
- Take the square root of the sum: \(\sqrt{25} = 5\)

Remember that the distance formula can be used in three dimensions by including the z-coordinates, transforming the formula to \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\].

## Euclidean Distance Formula

The Euclidean distance formula is a cornerstone in geometry. It allows you to find the shortest path between two points in a plane or space, given their coordinates.

### Derivation of the Formula

The formula for Euclidean distance in a two-dimensional space can be derived using the Pythagorean Theorem. For points \(A = (x_1, y_1)\) and \(B = (x_2, y_2)\), the horizontal and vertical distances between the points form the two legs of a right triangle, with the distance \(d\) being the hypotenuse.

In three dimensions, the Euclidean distance formula extends to include the z-coordinates as well. For points \(A = (x_1, y_1, z_1)\) and \(B = (x_2, y_2, z_2)\), the distance \(d\) is calculated as follows:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

### Step-by-Step Calculation

**Step 1:**Subtract the x-coordinates of the points to find the horizontal distance.**Step 2:**Subtract the y-coordinates of the points to find the vertical distance.**Step 3:**Square both distances.**Step 4:**Sum the squares of the distances.**Step 5:**Take the square root of the sum to get the Euclidean distance.

Consider two points, \(A = (1, 2)\) and \(B = (4, 6)\).Using the Euclidean distance formula:

- Subtract the x-coordinates: \(4 - 1 = 3\)
- Subtract the y-coordinates: \(6 - 2 = 4\)
- Square the differences: \(3^2 = 9\) and \(4^2 = 16\)
- Sum the squares: \(9 + 16 = 25\)
- Take the square root of the sum: \(\sqrt{25} = 5\)

Double-check your calculations to avoid common mistakes such as sign errors when subtracting coordinates.

### Applications of the Euclidean Distance Formula

The Euclidean distance formula is widely used in various fields:

- In computer science, for clustering algorithms like K-means.
- In physics, to calculate the distance between objects.
- In geography, to find the distance between two places on a map.

## Distance Formula Examples

Understanding how to apply the distance formula through examples can make the concept clearer and more intuitive. Let's delve into some practical scenarios to enhance your grasp of this important mathematical tool.

### Example with Coordinates in 2D Space

Consider points \(A = (2, 3)\) and \(B = (5, 7)\). The distance \(d\) is determined as follows:

**Step 1:**Subtract the x-coordinates: \(5 - 2 = 3\)**Step 2:**Subtract the y-coordinates: \(7 - 3 = 4\)**Step 3:**Square both differences: \(3^2 = 9\) and \(4^2 = 16\)**Step 4:**Add the squares: \(9 + 16 = 25\)**Step 5:**Take the square root: \(\sqrt{25} = 5\)

In higher dimensions, the distance formula adapts to include additional coordinates. For instance, in four-dimensional space for points \(A = (x_1, y_1, z_1, w_1)\) and \(B = (x_2, y_2, z_2, w_2)\), the formula becomes:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (w_2 - w_1)^2}\]. This adjustment allows for the computation of distances in even more complex spaces.

### Example with Real-World Application

Imagine you are calculating the distance between two points on a graph that represents a real-world scenario. Let's use the distance formula with specific coordinates to illustrate this.

You have two locations on a map, represented by the coordinates \(A = (1, 2)\) and \(B = (4, 6)\). To determine the distance between these points:

**Step 1:**Find the horizontal difference: \(4 - 1 = 3\)**Step 2:**Find the vertical difference: \(6 - 2 = 4\)**Step 3:**Square the differences: \(3^2 = 9\) and \(4^2 = 16\)**Step 4:**Add the squares: \(9 + 16 = 25\)**Step 5:**Take the square root: \(\sqrt{25} = 5\)

While using the distance formula, ensure you square the differences, not the coordinates themselves.

### Using the Distance Formula in a Cartesian Plane

The distance formula is frequently applied in Cartesian planes to find the shortest path between two points. This method is not just theoretical—it’s widely applicable in various practical fields.

Consider an example where two points are given: \(C = (6, 8)\) and \(D = (3, 4)\). You can find the distance \(d\) as follows:

**Step 1:**Calculate the x-coordinate difference: \(6 - 3 = 3\)**Step 2:**Calculate the y-coordinate difference: \(8 - 4 = 4\)**Step 3:**Square these differences: \(3^2 = 9\) and \(4^2 = 16\)**Step 4:**Add the squares: \(9 + 16 = 25\)**Step 5:**Take the square root of the sum: \(\sqrt{25} = 5\)

Consider Cartesian planes where axes aren't perpendicular, such as in certain trigonometric calculations or when non-Euclidean geometry is involved. The basic concepts of distance remain valid, but these alternate geometries require adaptations of the formula to accommodate curved spaces or different metrics.

## Application of Distance Formula

The distance formula is extensively used in various branches of mathematics, especially in analytic geometry, to find the distance between two points in a plane or space. Understanding this formula is essential for tackling numerous geometric problems and real-world scenarios.

### Distance Formula Explained

**The distance formula** is derived from the Pythagorean theorem. In a two-dimensional Cartesian plane, it calculates the distance between two points, helping you visualise their relative positions on the plane.The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Take points \(P = (2, 3)\) and \(Q = (5, 7)\). The distance \(d\) between these points can be calculated as follows:

- Subtract the x-coordinates: \(5 - 2 = 3\)
- Subtract the y-coordinates: \(7 - 3 = 4\)
- Square the differences: \(3^2 = 9\) and \(4^2 = 16\)
- Add the squares: \(9 + 16 = 25\)
- Take the square root: \(\sqrt{25} = 5\)

### Distance Formula Derivation

The distance formula originates from the Pythagorean theorem, which is applicable in right-angled triangles. By considering the distance between two points as the hypotenuse of a right-angled triangle, you can simplify the computation.For two points, \((x_1, y_1)\) and \((x_2, y_2)\) in a plane, the horizontal distance \((x_2 - x_1)\) and the vertical distance \((y_2 - y_1)\) form the two legs of the right triangle.

Using the Pythagorean theorem:\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.Applying to our coordinates:\[c = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This is how the distance formula is derived.

In three dimensions, the distance formula incorporates the z-coordinates, adding another term in the square root.

### Distance Formula in Coordinate Geometry

In coordinate geometry, the distance formula is vital for calculating the distance between points in a Cartesian plane. It simplifies the process of determining the length of line segments and the spatial relationships between points.

To find the distance between points \((3, 4)\) and \((7, 1)\) within a coordinate plane:

- Subtract x-coordinates: \(7 - 3 = 4\)
- Subtract y-coordinates: \(1 - 4 = -3\)
- Square these differences: \(4^2 = 16\) and \((-3)^2 = 9\)
- Add the squares: \(16 + 9 = 25\)
- Take the square root: \(\sqrt{25} = 5\)

In coordinate geometry, understanding the distance formula aids in solving more complex geometric problems, such as finding the equation of a circle, determining midpoints, and solving for areas of polygons. The formula remains effective across different dimensions and coordinates.

### Real-World Applications of Distance Formula

The distance formula is not confined to theoretical mathematics; it has practical applications across various fields. Here are some real-world scenarios where the distance formula plays a crucial role.

**Navigation:** Used in GPS technology to calculate the shortest path between two geographical locations.

In navigation, the distance formula helps in determining routes. For instance, calculating the shortest distance between two cities based on their coordinates to provide efficient travel routes.

**Engineering and Physics:** Utilised in calculating distances between objects, structural analysis, and motion paths.

In engineering, the distance formula is crucial for ensuring the proper alignment and placement of structures. In physics, it's essential for calculating the trajectory of particles, analysing forces, and understanding spatial relationships in various contexts.

**Computer Science:** Essential in algorithms such as K-means clustering, image processing, and network topology analysis.

The distance formula is adaptable to higher dimensions, meaning it can be used in three-dimensional space for even more versatile applications.

## Distance formula - Key takeaways

**Distance Formula Definition:**A mathematical tool used to calculate the distance between two points in a 2D or 3D space based on their Cartesian coordinates.**Distance Formula Explained:**Derived from the Pythagorean theorem, it computes the distance between points \(x_1, y_1\) and \(x_2, y_2\) using \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] for 2D space.**Distance Formula Derivation:**Based on the horizontal and vertical distances forming a right triangle, with the distance as the hypotenuse, leading to the simplified formula.**Euclidean Distance Formula:**An extension of the distance formula for 3D space incorporating z-coordinates: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\].**Applications of Distance Formula:**Utilised in fields like computer science (K-means clustering), physics (distance between objects), and geography (distances on maps).

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