About 3D Coordinates Calculator

What is the 3D Coordinates Calculator?

The 3D Coordinates Calculator is an advanced tool designed to determine the coordinates of a point in three-dimensional space. It calculates the position of a point that divides a line segment between two given points in a specified ratio. This calculator is particularly useful in 3D geometry, computer graphics, physics, and engineering applications where precise spatial positioning is crucial.

Formula

The formula to find the coordinates of a point P(x, y, z) that divides the line segment between points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio m:n is:

\[ x = x_1 + \frac{m}{m+n}(x_2 - x_1) \]

\[ y = y_1 + \frac{m}{m+n}(y_2 - y_1) \]

\[ z = z_1 + \frac{m}{m+n}(z_2 - z_1) \]

Where:

• (x₁, y₁, z₁) are the coordinates of the first point
• (x₂, y₂, z₂) are the coordinates of the second point
• m:n is the given ratio

Calculation Steps

1. Identify the coordinates of the two given points: (x₁, y₁, z₁) and (x₂, y₂, z₂)
2. Determine the ratio m:n
3. Calculate the ratio factor: r = m / (m + n)
4. Apply the formula for x-coordinate: x = x₁ + r(x₂ - x₁)
5. Apply the formula for y-coordinate: y = y₁ + r(y₂ - y₁)
6. Apply the formula for z-coordinate: z = z₁ + r(z₂ - z₁)
7. Round the results to the desired number of decimal places

Example and Visual Representation

Let's consider an example:

• Point 1: (1, 2, 3)
• Point 2: (7, 8, 9)
• Ratio: 2:3

Calculation:

1. Ratio factor: r = 2 / (2 + 3) = 0.4
2. x = 1 + 0.4(7 - 1) = 3.4
3. y = 2 + 0.4(8 - 2) = 4.4
4. z = 3 + 0.4(9 - 3) = 5.4

Therefore, the coordinates of point P are (3.4, 4.4, 5.4).

This example demonstrates how we can determine the coordinates of a point in 3D space using the given ratio. The visual representation helps to understand the position of the calculated point relative to the two given points in a three-dimensional context.