3D Geometry / Vector Lab
Real engineering & fabrication math: 3D distance, plane equations, line-plane intersection, vector angles, surface normals, rotation matrices, a quaternion calculator, coordinate transforms (Cartesian / spherical / cylindrical) and 2D/3D point rotation — with a live 3D view. For CAD, game dev, robotics and graphics.
How to Use
- Pick a tool, then type vectors and points as three numbers separated by spaces — e.g. <code>1 0 0</code>. Angles are in degrees; quaternions are <code>w x y z</code>.
- The <strong>Vector calculator</strong> gives dot, cross, magnitudes, angle, projection and unit vectors at once; the <strong>cross product</strong> is also your surface normal.
- <strong>Geometry</strong> tools: 3D distance (point–point, point–line, point–plane), the plane through three points (ax + by + cz + d = 0), and where a line pierces a plane.
- <strong>Rotation</strong> tools: build a rotation matrix from an axis & angle or Euler angles, rotate a point in 2D or 3D, and a full quaternion calculator (multiply, conjugate, to/from axis-angle, to matrix, rotate a vector).
- Vectors, normals and rotations are drawn in a small isometric 3D view so you can see the result; coordinate transforms convert between Cartesian, spherical and cylindrical.
3D view (isometric)
Formulas & conventions
About the 3D Geometry / Vector Lab
The 3D Geometry / Vector Lab is a simple, free helper for everyday maths and number work that runs entirely on your own device. Real engineering & fabrication math: 3D distance, plane equations, line-plane intersection, vector angles, surface normals, rotation matrices, a quaternion calculator, coordinate transforms (Cartesian / spherical / cylindrical) and 2D/3D point rotation — with a live 3D view. For CAD, game dev, robotics and graphics.
How it works
Enter your figures and the result appears instantly, updating the moment you change anything. There is no submit button and nothing to wait for, so it is easy to try a few what-if numbers and compare the results. Just check each box holds the kind of value it expects.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
What can it compute?
<strong>Vectors:</strong> dot & cross product, magnitude, angle between, projection, unit vectors. <strong>Geometry:</strong> point-to-point / point-to-line / point-to-plane distance, the plane equation through three points (with its normal and the triangle area), surface normals, and line-plane intersection. <strong>Rotations:</strong> rotation matrices from axis-angle or Euler angles, 2D and 3D point rotation, and a quaternion calculator. <strong>Transforms:</strong> Cartesian ↔ spherical ↔ cylindrical.
Why use quaternions instead of Euler angles?
Quaternions represent 3-D rotations without <em>gimbal lock</em> (the loss of a degree of freedom that Euler angles suffer near 90°), they interpolate smoothly (slerp), and composing rotations is a single multiplication. That's why game engines, robotics and spacecraft attitude systems use them. This tool converts between quaternions, axis-angle, rotation matrices and Euler angles so you can move between representations.
How is the plane equation found?
From three points it computes two edge vectors, takes their cross product to get the normal (a, b, c), and sets d = −(normal · point), giving ax + by + cz + d = 0. The cross product's length is twice the triangle's area, so you also get the area and the unit surface normal — exactly what you need for shading or toolpath orientation.
What conventions does it use?
Right-handed coordinates. Spherical is (r, θ, φ) with θ the inclination from +z and φ the azimuth from +x. Euler angles are applied in XYZ order (R = Rz·Ry·Rx — rotate about X, then Y, then Z). Quaternions are stored as [w, x, y, z]. These match the most common CAD/graphics conventions; the output labels each one.
What is line-plane intersection good for?
Finding where a ray meets a surface — the core of ray casting and ray tracing, click-to-select in 3-D viewers, collision and projection in CAD, and computing where a robot's tool axis meets a work plane. The tool returns the intersection point and the line parameter t, or tells you the line is parallel to (or lies in) the plane.
Does it run locally?
Yes — all the vector, matrix and quaternion math and the 3-D view run in your browser. Nothing is uploaded and it works offline.
How do I use the 3D Geometry / Vector Lab?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
CAD & fabrication
Plane equations, surface normals and distances for toolpaths and fixtures.
Game development
Rotation matrices, quaternions and ray-plane hits for movement and picking.
Robotics
Pose math, axis-angle ↔ quaternion ↔ Euler, and coordinate frames.
Computer graphics
Normals for shading, transforms between spaces, vector projections.
3D printing / CNC
Angles between faces, point rotation, distances and orientation.
Engineering & physics
Resolve forces, find angles, and convert coordinate systems.
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