Result Summary
The calculated line equations appear here after submission.
Calculator Inputs
Use either two known points or one point with a direction vector.
3D Plotly Graph
The graph shows the line path, starting point, reference point, evaluated point, and closest point to the origin.
Example Data Table
Sample example with two known points| Mode | Point P1 | Point P2 / Direction | t | Direction Vector | Vector Form | Point at t |
|---|---|---|---|---|---|---|
| Two points | (1, 2, 3) | (5, 6, 9) | 0.5 | <4, 4, 6> | r = <1, 2, 3> + t<4, 4, 6> |
(3, 4, 6) |
Formula Used
For a point P(x1, y1, z1) and direction vector d = <a, b, c>, the line can be written in three standard ways.
Vector form
r = <x1, y1, z1> + t<a, b, c>
Parametric form
x = x1 + aty = y1 + btz = z1 + ct
Symmetric form
(x - x1)/a = (y - y1)/b = (z - z1)/c
If any direction component is zero, that axis stays constant.
Two-point direction vector
d = <x2 - x1, y2 - y1, z2 - z1>
Point on the line at parameter t
P(t) = (x1 + at, y1 + bt, z1 + ct)
How to Use This Calculator
- Select Two points if you know both coordinates on the same line.
- Select Point and direction vector if you know one point and the line direction.
- Enter the parameter value t to evaluate a specific point.
- Set a minimum and maximum t range for the 3D graph.
- Press Calculate 3D Line to show the result above the form.
- Use Download CSV for spreadsheet data or Download PDF for a printable summary.
Frequently Asked Questions
1. What inputs can this calculator accept?
It accepts either two points in 3D space or one point together with a direction vector. Both methods generate the same line forms and supporting metrics.
2. What does the parameter t mean?
The parameter t moves along the line. When t changes, the calculator substitutes it into the parametric equations and returns the exact coordinate on that line.
3. Why might the symmetric form include a constant axis?
If one direction component is zero, that coordinate never changes along the line. The calculator keeps that axis constant instead of dividing by zero.
4. Can I use decimal or negative values?
Yes. The calculator accepts integers, decimals, and negative values for points, direction components, and graph ranges, which is useful for analytic geometry work.
5. What does the distance from origin show?
It shows the shortest perpendicular distance from the origin to the line. This helps when studying geometric position and comparing nearby lines in space.
6. What are the axis angles used for?
The axis angles describe the line direction relative to the x, y, and z axes. They come from the direction cosines of the line vector.
7. Does the graph represent the full infinite line?
The line is mathematically infinite, but the graph shows a chosen segment using your minimum and maximum t values. Increase the range to view more.
8. When will the calculator reject my input?
It rejects identical points or a zero direction vector because those do not define a unique 3D line. It also checks that the graph range is valid.