3D Section Formula Calculator

This 3D Section Formula Calculator helps you find the coordinates of the point that divides the line segment between two points in three-dimensional space in a given ratio. It can be used for both internal division and external division. Substitute the coordinates and ratio into the correct 3D section formula, then simplify each coordinate to get the final point. It is a simple way to check answers, understand the formula clearly, and practise coordinate geometry step by step.

Step-by-step method

1. Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.
2. Write the correct 3D section formula.
3. Substitute the values into the formula.
4. Simplify to get the final dividing point.

Formula:

Example 1:

Step 1 - Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.

In this problem: The points are \(P_{1} = \left(1,\; 2,\; 3\right)\) and \(P_{2} = \left(4,\; 6,\; 3\right)\), divided in the ratio \(2 : 1\) (internal division).

Step 2 - Write the correct 3D section formula.

In this problem: Use the section formula with the sign that matches the division type.

Step 3 - Substitute the values into the formula.

In this problem: Substitute the coordinates and the ratio.

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(3,\; \frac{14}{3},\; 3\right)\).

Final answer:

Example 2:

Step 1 - Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.

In this problem: The points are \(P_{1} = \left(1,\; 0,\; 2\right)\) and \(P_{2} = \left(5,\; 4,\; 6\right)\), divided in the ratio \(3 : 1\) (external division).

Step 2 - Write the correct 3D section formula.

In this problem: Use the section formula with the sign that matches the division type.

Step 3 - Substitute the values into the formula.

In this problem: Substitute the coordinates and the ratio.

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(7,\; 6,\; 8\right)\).

Final answer:

See Example 1 Hide Example 1

Problem

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(4,\; 6,\; 3\right),\; m : n = 2 : 1\; (\text{internal})\)

Approach

Step-by-step method

1. Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.
2. Write the correct 3D section formula.
3. Substitute the values into the formula.
4. Simplify to get the final dividing point.

Step 1

Step 1 - Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.

In this problem: The points are \(P_{1} = \left(1,\; 2,\; 3\right)\) and \(P_{2} = \left(4,\; 6,\; 3\right)\), divided in the ratio \(2 : 1\) (internal division).

\(P_{1} = \left(1,\; 2,\; 3\right),\; P_{2} = \left(4,\; 6,\; 3\right),\; m : n = 2 : 1\; (\text{internal})\)

Step 2

Step 2 - Write the correct 3D section formula.

In this problem: Use the section formula with the sign that matches the division type.

\(\text{Internal: } P = \left(\frac{m x_{2} + n x_{1}}{m + n},\; \frac{m y_{2} + n y_{1}}{m + n},\; \frac{m z_{2} + n z_{1}}{m + n}\right)\quad\text{External: replace } + \text{ with } -\)

Step 3

Step 3 - Substitute the values into the formula.

In this problem: Substitute the coordinates and the ratio.

\(P = \left(\frac{2 \cdot 4 + 1 \cdot 1}{2 + 1},\; \frac{2 \cdot 6 + 1 \cdot 2}{2 + 1},\; \frac{2 \cdot 3 + 1 \cdot 3}{2 + 1}\right)\)

Step 4

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(3,\; \frac{14}{3},\; 3\right)\).

\(P = \left(3,\; \frac{14}{3},\; 3\right)\)

Final Answer

\(P = \left(3,\; \frac{14}{3},\; 3\right)\)

See Example 2 Hide Example 2

Problem

\(P_{1} = \left(1,\; 0,\; 2\right),\; P_{2} = \left(5,\; 4,\; 6\right),\; m : n = 3 : 1\; (\text{external})\)

Approach

Step-by-step method

1. Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.
2. Write the correct 3D section formula.
3. Substitute the values into the formula.
4. Simplify to get the final dividing point.

Step 1

Step 1 - Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.

In this problem: The points are \(P_{1} = \left(1,\; 0,\; 2\right)\) and \(P_{2} = \left(5,\; 4,\; 6\right)\), divided in the ratio \(3 : 1\) (external division).

\(P_{1} = \left(1,\; 0,\; 2\right),\; P_{2} = \left(5,\; 4,\; 6\right),\; m : n = 3 : 1\; (\text{external})\)

Step 2

Step 2 - Write the correct 3D section formula.

In this problem: Use the section formula with the sign that matches the division type.

\(\text{Internal: } P = \left(\frac{m x_{2} + n x_{1}}{m + n},\; \frac{m y_{2} + n y_{1}}{m + n},\; \frac{m z_{2} + n z_{1}}{m + n}\right)\quad\text{External: replace } + \text{ with } -\)

Step 3

Step 3 - Substitute the values into the formula.

In this problem: Substitute the coordinates and the ratio.

\(P = \left(\frac{3 \cdot 5 - 1 \cdot 1}{3 - 1},\; \frac{3 \cdot 4 - 1 \cdot 0}{3 - 1},\; \frac{3 \cdot 6 - 1 \cdot 2}{3 - 1}\right)\)

Step 4

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(7,\; 6,\; 8\right)\).

\(P = \left(7,\; 6,\; 8\right)\)

Final Answer

\(P = \left(7,\; 6,\; 8\right)\)