What breaks first when you trust θ blindly?
Angle mode divides by both lengths. The moment either vector collapses to (0,0,0), inverse cosine has nothing stable to bite. The tool stops with a short warning instead of printing nonsense degrees.
Floating noise matters near parallel directions. We clamp cosine to [-1,1] before arccos so rounding error does not throw NaN. You still should treat 179.999° style outputs as “almost opposite,” not a license to ignore measurement error in real sensors.
Operation snapshot
Each chip swaps the algebra underneath while keeping the same lanes. Use the table as a cheat sheet when you forget whether the result stays scalar or grows a new arrow.
| Mode | Output type | Primary formula hint |
|---|---|---|
| Dot | Scalar | Sum of pairwise products axbx + ayby + azbz |
| Cross | Vector in ℝ³ | Determinant layout with unit axes |
| Lengths | Two scalars | Euclidean √(x² + y² + z²) per lane |
| Angle | Degrees and radians | arccos of dot product over product of lengths |
| A + B | Vector | Add each matching component |
| A − B | Vector | Subtract each matching component |
How values move through this page
You type decimals into six fields. On each keystroke the script reads both vectors, recomputes the active operation, injects structured HTML into the readout panel, then rescales the canvas so arrows stay inside the rounded frame. ResizeObserver watches the wrapper width so phones in portrait and ultrawide desktops share one code path.
We treat empty or non-numeric input as zero. Fast for homework, slightly blunt for symbolic work. If you need strict validation banners, pair this page with pencil checks or a CAS you already trust.
Three workloads people paste in here
- Physics homework on force components
- Students set A as a force vector, B as a displacement or second force, then flip between dot and cross to match the chapter diagram. The xy preview helps sanity-check whether they flipped a sign on east vs north.
- Graphics debugging before pushing shaders
- TDs normalize rough camera-right vectors against up vectors, then inspect θ to see whether orthogonality drifted after a bad matrix import. They still verify in-engine, yet the quick numeric pass catches typos early.
- Robotics teaching labs
- Instructors demo torque intuition by showing |A × B| grow as inputs leave the plane even while the 2D plot stays flat. The contrast sparks questions instead of hiding z entirely.
When you only need the scalar alignment score between two lists of three numbers, the dedicated dot product calculator page trims the interface. Stay here when you want cross, subtraction, and length in one pass.
Right-hand rule without the chalk dust
Cross output points along the axis where your right-hand fingers curl from A toward B through the smaller angle. The tool does not render a 3D arrow for that normal, so you read the numeric triple and imagine the corkscrew yourself. If your physics class uses left-handed frames, flip one input’s sign on purpose and compare against the textbook diagram before you blame the software.
Game engines disagree on column vs row vectors and on whether Z grows up or into the screen. This page assumes plain Cartesian triples you type yourself. Encode your studio’s handedness by flipping signs yourself instead of hunting for a buried preset.
Where batch work belongs elsewhere
Spreadsheet columns with ten thousand rows deserve a dedicated compute environment. Here you get transparent formulas in the readout for one pair at a time, which keeps the mental model obvious during grading or code review. The moment you need matrix-vector products across a whole mesh, export components and continue inside numpy, Julia, or the tool your team already standardized.
Symbolic simplification is another boundary. If you must prove orthogonality under parameters named t, a computer algebra system keeps track of cancellations. We stay numeric on purpose so phones load fast and instructors demo without installing runtimes.
Judgment call on precision
Four decimal places keep the readout readable on small screens. It is not IEEE benchmark territory. For publication-grade error bars, export components to your usual spreadsheet or numpy stack.
Units live in your head here
The inputs are raw numbers. If A is Newtons and B is meters, the dot lands in joules once you carry the unit algebra mentally. If both are dimensionless feature vectors, the dot is a similarity score without extra scaling. The software never invents radians where you meant degrees; angle mode always reports both units side by side so you spot a mismatch before tweeting a screenshot.
Teachers sometimes ask whether rounding should happen before or after squaring for length. We square first at full float precision inside the browser, then round for display. Repeat the same numbers on paper if a rubric demands intermediate rounding, because pedagogy occasionally disagrees with IEEE defaults.
