Calculator Form
Choose a mode, enter coordinates, and submit. The result appears above this form, directly below the page header.
Example Data Table
| Case | Inputs | Output | Main Result |
|---|---|---|---|
| Point from origin | P = (3, 4, 12) | OP = <3, 4, 12> | |OP| = 13 |
| Two-point vector | A = (1, 2, 0), B = (5, 6, 0) | AB = <4, 4, 0> | |AB| = 5.656854 |
| Midpoint | A = (2, -1, 3), B = (6, 5, 9) | M = (4, 2, 6) | OM = <4, 2, 6> |
| Internal section | A = (0, 0, 0), B = (10, 5, 0), ratio 2:3 | R = (4, 2, 0) | OR = <4, 2, 0> |
Formula Used
1) Position vector of a point
For point P(x, y, z), the position vector from the origin is:
OP = xi + yj + zk
2) Vector between two points
For A(x₁, y₁, z₁) and B(x₂, y₂, z₂):
AB = (x₂ − x₁)i + (y₂ − y₁)j + (z₂ − z₁)k
3) Magnitude of a vector
|r| = √(x² + y² + z²)
4) Unit vector
r̂ = r / |r|, valid only when the vector is not zero.
5) Midpoint formula
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
6) Internal section formula
R = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))
7) External section formula
R = ((mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n), (mz₂ − nz₁)/(m−n))
How to Use This Calculator
- Select the calculation mode based on the problem you want to solve.
- Choose 2D for x-y work or 3D for x-y-z work.
- Enter Point P for a direct position vector, or Point A and Point B for two-point calculations.
- For section problems, enter ratio values m and n, then choose internal or external division.
- Press the calculate button to show the full result above the form.
- Review the vector notation, basis form, magnitude, unit vector, direction data, result table, and graph.
- Use the CSV or PDF buttons to export the computed results.
FAQs
1) What is a position vector?
A position vector starts at the origin and ends at a point. Its components show exactly where that point lies in the coordinate system.
2) Can this calculator solve both 2D and 3D problems?
Yes. Choose 2D for plane geometry or 3D for spatial geometry. The calculator automatically adjusts formulas, fields, and graph type.
3) How is the vector between two points found?
Subtract each coordinate of the starting point from the corresponding coordinate of the ending point. That gives the displacement vector from A to B.
4) What does the midpoint mode return?
It finds the point halfway between A and B, then expresses that midpoint as a position vector measured from the origin.
5) What is the difference between internal and external division?
Internal division places the section point between A and B. External division places it on the extended line outside the segment.
6) Why can the unit vector become undefined?
A unit vector requires division by the vector magnitude. When the magnitude is zero, division is impossible, so the unit vector is undefined.
7) What do direction cosines and direction angles mean?
Direction cosines describe how a vector aligns with coordinate axes. Direction angles are the corresponding angles with those axes.
8) Why does the graph sometimes change style?
The plot switches automatically. Two-dimensional inputs use a flat coordinate graph, while three-dimensional inputs use a 3D spatial graph.