3D Section Formula Notes

Published on: September 29, 2026

This 3D Section Formula Calculator finds the point that divides a segment in a given ratio, and shows the working line by line. For an internal division the point sits between the two ends, and for an external one it sits beyond them, which is the only difference the sign in the formula makes. Each coordinate is worked out on its own. Following the same pattern every time keeps the ratio the right way round.

Step-by-step method

Set up the problem.
Write the correct 3D section formula.
Substitute the values into the formula.
Simplify to get the final dividing point.

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

Set up the problem.
Write the correct 3D section formula.
Substitute the values into the formula.
Simplify to get the final dividing point.

Step 1

Step 1 - Set up the problem.

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

Set up the problem.
Write the correct 3D section formula.
Substitute the values into the formula.
Simplify to get the final dividing point.

Step 1

Step 1 - Set up the problem.

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

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

See Example 3 Hide Example 3

Problem

\(P_{1} = \left(2,\; -1,\; 4\right),\; P_{2} = \left(8,\; 5,\; 1\right),\; m : n = 1 : 2\; (\text{internal})\)

Approach

Step-by-step method

Set up the problem.
Write the correct 3D section formula.
Substitute the values into the formula.
Simplify to get the final dividing point.

Step 1

Step 1 - Set up the problem.

In this problem: The points are \(P_{1} = \left(2,\; -1,\; 4\right)\) and \(P_{2} = \left(8,\; 5,\; 1\right)\), divided in the ratio \(1 : 2\) (internal division).

\(P_{1} = \left(2,\; -1,\; 4\right),\; P_{2} = \left(8,\; 5,\; 1\right),\; m : n = 1 : 2\; (\text{internal})\)

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.

\(P = \left(\frac{1 \cdot 8 + 2 \cdot 2}{1 + 2},\; \frac{1 \cdot 5 + 2 \cdot \left(-1\right)}{1 + 2},\; \frac{1 \cdot 1 + 2 \cdot 4}{1 + 2}\right)\)

Step 4

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(4,\; 1,\; 3\right)\).

\(P = \left(4,\; 1,\; 3\right)\)

Final Answer

\(P = \left(4,\; 1,\; 3\right)\)

See Example 4 Hide Example 4

Problem

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

Approach

Step-by-step method

Set up the problem.
Write the correct 3D section formula.
Substitute the values into the formula.
Simplify to get the final dividing point.

Step 1

Step 1 - Set up the problem.

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

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

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.

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

Step 4

Step 4 - Simplify to get the final dividing point.

In this problem: The dividing point is \(\left(8,\; 10,\; 9\right)\).

\(P = \left(8,\; 10,\; 9\right)\)

Final Answer

\(P = \left(8,\; 10,\; 9\right)\)